Computational geometry using control geometry having at least two dimensions

ABSTRACT

A method and system for computer aided design (CAD) is disclosed for designing geometric objects. The present invention interpolates and/or blends between such geometric objects sufficiently fast so that real time deformation of such objects occurs while deformation data is being input. Thus, a user designing with the present invention obtains immediate feedback to input modifications without separately entering a command for performing such deformations. The present invention utilizes novel computational techniques for blending between geometric objects, wherein weighted sums of points on the geometric objects are used in deriving a new blended geometric object. The present invention is particularly useful for designing the shape of surfaces. Thus, the present invention is applicable to various design domains such as the design of, e.g., bottles, vehicles, and watercraft. Additionally, the present invention provides for efficient animation via repeatedly modifying surfaces of an animated object such as a representation of a face.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application is a divisional of U.S. patent application Ser.No. 09/360,029 filed Jul. 23,1999, which claims priority from U.S.Provisional Application Ser. No. 60/093,892, filed Jul. 23, 1998, andfrom U.S. Provisional Application Ser. No. 60/116,199, filed Jan. 15,1999, all of which are incorporated herein by reference in theirentirety.

FIELD OF THE INVENTION

The present invention relates to a system and method for performingcomputer aided design, and, in particular, to efficient computationaltechniques for blending between representations of geometric objects.

BACKGROUND

A designer using a computer aided design (CAD) computational system willtypically approach the design of a free form geometric object (such as asurface) by first specifying prominent and/or necessary subportions ofthe geometric object through which the object is constrained to pass.Subsequently, a process is activated for generating the geometric objectthat conforms to the constraining subportions provided. In particular,such subportions may be points, curves, surfaces and/or higherdimensional geometric objects. For example, a designer that designs asurface may construct and position a plurality of curves through whichthe intended surface must pass (each such curve also being denotedherein as a “feature line” or “feature curve”). Thus, the intendedsurface is, in general, expected to have geometric characteristics (suchas differentiability and curvature) that, substantially, only change tothe extent necessary in order to satisfy the constraints placed upon thesurface by the plurality of curves. That is, the designer expects thegenerated surface to be what is typically referred to as “fair” by thoseskilled in the art. Thus, the designer typically constructs such featurecurves and positions them where the intended surface is likely to changeits geometric shape in a way that cannot be easily interpolated fromother subportions of the surface already designed.

As a more specific example, when designing containers such as bottles,an intended exterior surface of a bottle may be initially specified bysubportions such as: (a) feature curves positioned in high curvatureportions of the bottle surface, and (b) surface subareas havingparticular geometric characteristics such as having a shape or contourupon which a bottle label can be smoothly applied. Thus, the intentionof a bottle surface designer is to construct a bottle design thatsatisfies his/her input constraints and that is also fair. Moreover, thedesigner may desire to generate holes for handles, as well as, e.g.,ergonomic bottle grips by deforming various portions of the bottlesurface and still have the bottle surface fair.

There has heretofore, however, been no CAD system wherein a designer (ormore generally, user) of geometric objects can easily and efficientlyexpress his/her design intent by inputting constraints and having theresulting geometric object be fair. That is, the designer/user mayencounter lengthy delays due to substantial computational overheadand/or the designer/user may be confronted with non-intuitive geometricobject definition and deformation techniques that require substantialexperience to effectively use. For example, many prior art CAD systemsprovide techniques for allowing surfaces to be designed and/or deformedby defining and/or manipulating designated points denoted as “controlpoints.” However, such techniques can be computationally expensive,non-intuitive, and incapable of easily deforming more than a local areaof the surface associated with such a control point. Additionally, someprior art CAD systems provide techniques for defining and/or deformingsurfaces via certain individually designated control vectors. That is,the direction of these vectors may be used to define the shape orcontour of an associated surface. However, a designer's intent may noteasily correspond to a surface design technique using such controlvectors since each of the control vectors typically corresponds to onlya single point of the surface isolated from other surface points havingcorresponding control vectors. Thus, such techniques are, at most, onlyable to deform an area of the surface local to such points havingcorresponding control vectors.

Additionally, such prior art CAD systems may also have difficulties inprecisely performing blending and trimming operations. For example, twogeometric objects intended to abut one another along a common boundarymay not be within a sufficient tolerance to one another at the boundary.That is, there may be sufficiently large gaps between the geometricobjects that the boundary may not be considered “water tight,” which maybe problematic in certain machining operations and other operations likeBoolean operations on solids.

Accordingly, it would be very desirable to have a CAD system thatincludes one or more geometric design techniques for allowing CADdesigners/users to more easily, efficiently and precisely designgeometric objects. Further, it would be desirable to have such a systemand/or computational techniques for graphically displaying geometricobjects, wherein there is greater user control over the defining and/ordeforming of computational geometric objects, and in particular, moreintuitive global control over the shape or contour of computationallydesigned geometric objects.

Definitions

This section provides some of the fundamental definitions that are usedin describing the present invention. These definitions are alsoillustrated in FIGS. 15 and 16.

A “parametric geometric object” S is a geometric object that is theimage of a function f, wherein the domain of f is in a geometric spaceembedded within a coordinate system (denoted the “parameter space”) andthe range of f is in another geometric space (denoted the “objectspace”). Typically, the inverse or pre-image, f⁻¹, of a geometric objectsuch as S will be a geometrically simpler object than its image inobject space. For example, the pre-image of a curve 170 in object spacemay be a simple line segment 172, L, in parameter space. Thus if Sdenotes the curve in object space, then notationally f and S aresometimes identified such that for uεL, a corresponding point in thecurve S is denoted S(u). Similarly, the pre-image of an undulatingsurface 204 (FIG. 16) in object space may be a simple bounded plane 180in parameter space. Thus, if S denotes the undulating surface 204, thenfor (u,v)εf⁻¹(S), S(u,v) denotes a corresponding point on the undulatingsurface 204.

A “profile” 200 (FIG. 16) is a geometric object, such as a curve inobject space, through which an associated object space geometricallymodeled object (e.g. surface 204) must pass. That is, such profiles 200are used to generate the geometrically modeled object. Thus, profilesprovide a common and natural way for artists and designers togeometrically design objects, in that such a designer may think in termsof the feature or profile curves when defining the characteristic shapeof a geometric object (surface) being designed. For example, profilecurves on a surface may substantially define the geometry of a resultingderived geometric object; e.g., its continuity, curvature, shape,boundaries, kinks, etc. Note, that for many design applications,profiles are typically continuous and differentiable. However, suchconstraints are not necessary. For example, a profile may, in additionto supplying a general shape or trend of the geometric object passingtherethrough, also provide a texture to the surface of the geometricobject. Thus, if a profile is a fractal or fractal-like, the fractalcontours may be in some measure imparted to the surface of the derivedgeometric object adjacent the profile. Further note that it is withinthe scope of the present invention to utilize profiles that are ofhigher dimension (≧2). Thus, a profile may also be a surface or a solid.Accordingly, if a profile is a surface, then a solid having locally(i.e., adjacent to the profile) at least some of the geometriccharacteristics of the profile may be derived.

Moreover, profiles (and/or segments thereof) may have variouscomputational representations such as linear (e.g., hyperplanes),elliptic, NURBS, or Bezier. Note, however, that regardless of thecomputational representation, a method (such as interpolation) fordeforming or reshaping each profile is preferable. More particularly, itmay be preferable that such a method results in the profile satisfyingcertain geometric constraints such as passing through (or substantiallyso) one or more predetermined points, being continuous, beingdifferentiable, having a minimal curvature, etc. Further, note that sucha deformation method may also include the ability to decompose a profileinto subprofiles, wherein the common boundary (e.g., a point) betweenthe subprofiles may be “slidable” along the extent of the originalprofile.

A “marker” 208 (FIG. 16) is a point on a profile that can be moved tochange the shape of the profile 200 in a region about the marker. Amarker also designates a position on a profile where the shape of ageometric object having the profile thereon can be deformed.

A “profile handle” 212 (FIG. 16) is a geometric object tangent to theprofile 200. Such a profile handle may control the shape of the profilelocally by modifying the slope (derivative) of the profile at the marker208. Alternatively, for non-differentiable profiles, a profile handlemay be used to control the general shape of the profile by indicating atrend direction and magnitude of the corresponding profile. For example,if the profile is a fractal or other non-differentiable geometricobject, then a profile handle may, for example, provide a range withinthe object space to which the profile must be confined; i.e., the rangemay be of a tubular configuration wherein the profile is confined to theinterior of the tubular configuration, Note that the profile handle 212affects the fullness of the profile 200 (e.g., the degree of convexitydeviating from a straight line between markers on the profile) bychanging the length of the profile handle.

An “isocline boundary” 220 is the boundary curve opposite the profile200 on the isocline ribbon 216. In one embodiment, at each point on theprofile 200 there is a paired corresponding point on the isoclineboundary 200, wherein each such pair of points defines a vector 224(denoted a “picket”) that is typically transverse to a tangent vector atthe point on the profile. More particularly, for a parameterizedprofile, the isocline boundary 220 can be viewed as a collection ofpickets at all possible parameter values for the profile 200.

An “isocline ribbon” (or simply isocline) is a geometric object, such asa surface 216, which defines the slope of the geometric object (e.g.,surface) 204 (more generally a geometric object 204) at the profile 200.Equivalently, the isocline ribbon may be considered as therepresentation of a geometric object delimited by the profile 200, theisocline handles 218 a and 218 b (discussed hereinbelow), and theisocline boundary 220. In other words, the geometric object 204 must“heel” to the isocline ribbon 216 along the profile 200. Said anotherway, in one embodiment, the geometric object 204 must be continuous atthe isocline 216 and also be continuously differentiable across theprofile 200. In an alternative embodiment, the geometric object 204 maybe constrained by the isocline 216 so that the object 204 lies within aparticular geometric range in a similar manner as discussed above in thedescription of the term “profile.” Note that there may be two isoclineribbons 216 associated with each profile 200. In particular, for aprofile that is a boundary for two abutting surfaces (e.g. two abuttingsurfaces 204), there can be an isocline ribbon along the profile foreach of the two surfaces. Thus we may speak of a right and a left handisocline ribbon.

An “isocline handle” 228 is a geometric object (e.g., a vector) forcontrolling the shape of the isocline ribbon 216 at the marker 208,wherein the profile handle and isocline handle at the marker may definea plane tangent to the surface 204. Hence the isocline handle is used todetermine the shape of the surface 204 (or other underlying geometricobject) about the marker. In particular, an isocline handle 228 is auser manipulatible picket 224. If all the profile handles 212 andisocline handles 228 (e.g., for two or more abutting surfaces) arecoplanar at a marker 208, then the surface 204 will be smooth at themarker (assuming the surface is continuously differentiable), otherwisethe surface may have a crease or dart. Note that by pulling one of thehandles (either isocline or profile) out of the plane of the otherhandles at a marker, one may intentionally generate a crease in thesurface 204 along the profile 200.

The part of the profile 200 between two markers 208 is denoted a“profile segment” 232. Similarly, the part of the isocline ribbon 216between two isocline handles 228 is denoted a ribbon segment 240.

A “boundary segment” 244 denotes the part of the boundary 220 betweentwo isocline handles 228.

The vector 246 that is the derivative tangent to the isocline boundary220 at an isocline handle 228 is denoted a “ribbon tangent.” Note thatmodifications of ribbon tangents can also be used by the presentinvention to control and/or modify the shape of an underlying geometricobject such as surface 204.

Isocline handles 228 may be generalized to also specify curvature of thesurface 204. That is, instead of straight vectors as isocline handles,the handles may be curved and denoted as “isocline ribs” 248. Thus, suchribs may facilitate preserving curvature continuity between surfaceshaving associated isocline ribbons along a common profile boundary,wherein the isocline ribbons are composed of isocline ribs. Accordingly,the curvature of such surfaces will match the curvature of theircorresponding isocline ribs, in much the same way as they match intangency.

A “developable surface” is a surface that can be conceptually rolled outflat without tearing or kinking. It is a special case of a “ruledsurface,” this latter surface being defined by being able to lay a ruler(i.e., straight edge) at any point on the surface and find anorientation so that the ruler touches the surface along the entirety ofthe ruler. For a developable surface, the surface perpendiculars are allequal in direction along the ruling.

“Label surfaces” denote special 2-dimensional (developable or nearlydevelopable) surfaces wherein a label may be applied on, e.g., acontainer. Label surfaces allow application of a decal without tearingor creasing. These surfaces are highly constrained and are not typicallydeformed by the geometric modification of an isocline ribbon 216.

A “trim profile” is a geometric object (curve) that is a profile fortrimming another geometric object (e.g., a surface). The trim profilemay have a single corresponding isocline ribbon 216 since if the surfaceto be trimmed is a label surface, it will not be modified and,accordingly, no isocline ribbon can be used to change its shape.

A trim profile (curve) can be used to delimit any surface, not just alabel surface. A surface, S, that is blended along a trim profile withone or more other surfaces is called an “overbuilt surface” when thesurface S overhangs the trim profile. For example, in FIG. 12, surface130 is an overbuilt surface, wherein the portion of the surface outsideof the area 134 is typically not shown to the designer once it has beentrimmed away.

A convex combination of arguments F_(i) is a summation

$\sum\limits_{i}{c_{i}F_{i}}$where the c_(i) are scalar coefficients and scalar multiplication iswell-defined for the F_(i) (e.g., F_(i) being vectors, functions, ordifferential operators), and where c_(i)≧0 and

${\sum\limits_{i}c_{i}} = 1.$If the F_(i) are points in space, for instance, then the set of allpossible such combinations yields the convex hull of the points F_(i),as one skilled in the art will understand.

A “forward evaluation” is a geometric object evaluation technique,wherein in order to generate a set of sample values from a function,f(x), argument values for x are incremented and f is subsequentlyevaluated. This type of evaluation is usually fast and efficient, butdoes not give function values at chosen positions between theincrements.

An “implicit function” is one written in the form f(x)=0. XεR^(N) When aparametric curve or surface is converted to an implicit form, theconversion is called “implicitization.” Hence f(t)=(sin(t), cos(t)) inparametric form may be implicitized by f(x,y)=x²+y²−1=0. Both formsdescribe a circle.

Dividing a vector by its length “normalizes” it. The normalized vectorthen has unit length. A vector function may be divided by its gradient,which will approximate unit length, as one skilled in the art willunderstand.

Given a function defined by a

$\sum\limits_{i}{{p_{i}(t)}{F_{i}(t)}}$where p_(i)(t) are weighting functions, if

${\sum\limits_{i}{p_{i}(t)}} = 1$for all values of t, then the p_(i) are said to form a “partition ofunity.”

“GI” continuity denotes herein a geometric continuity condition whereindirection vectors along a continuous parametric path on a parametricallydefined geometric object are continuous, e.g., tangent vector magnitudesare not considered.

SUMMARY OF THE INVENTION

The present invention is a computational geometric design system that iscapable of sufficiently efficient computations so as to allow real-timedeformations to objects such as surfaces while a user is supplying theobject modifying input. Thus, the present invention is a paradigm shiftaway from typical CAD systems since, in a typical CAD system the usermust supply input for changing or modifying a geometric object andsubsequently explicitly request processing of the input to commence.Thus, in such prior art CAD systems, the user receives feedback abouthis/her design at discrete user requested times. Instead, with thepresent invention, updates may be processed in real-time immediatelyupon input receipt without the user explicitly indicating that updateprocessing is to be performed.

Given the enhanced computational efficiency of the present invention, auser of the present invention can more efficiently perform iterativeapproximations to a geometric object being designed. The user mayspeedily design without the need to precisely calculate design geometriccharacteristics for portions of the object where such precision may notbe necessary. That is, the user can be less concerned about getting it“right the first time” since the ease of modification and speed ofcomputing modifications allows the user to more rapidly approximateand/or prototype a geometric object. Thus, the present invention canhave substantial efficiency benefits in that for many geometricallydesigned objects (including machined parts), substantial portions ofsuch objects may be satisfactorily designed with a wide range ofgeometric characteristics.

The CAD system of the present invention enables novel design techniquesby providing a novel computational technique for blending between twoparametric geometric objects such as surfaces. In one embodiment of thepresent invention, this novel blending technique blends between twoparametric surfaces S₁(u,v) and S₂(u,v), wherein each surface has, e.g.,the unit square as its parameter space. Assuming each surface S₁ and S₂has a respective blending function B₁(u,v) and B₂(u,v) such that each ofthe blending functions has, for example, (0,1) as its range for u and v(as well as satisfying other properties given hereinbelow), a newsurface, S, may be defined by the following formula:S(u,v)=S ₁(u,v)·B ₁(u,v)+S ₂(u,v)·B ₂(u,v)  (1)Note that the blending functions B₁ and B₂ are typically chosen so thatthe resulting blended surface S is the same as S₁ on a boundary with S₁,and the same as S₂ on a boundary with S₂. This is achieved by devisingB₁ and B₂ so that B₁=1 and B₂=0 on the boundary with S₁ while havingB₂=1 and B₁=0 on the boundary with S₂.

In a more general embodiment, the present invention may be used forblending between a plurality of geometric objects, S_(i), i=1, . . . ,N, wherein each of the geometric objects is parameterized by acorresponding function f_(s) _(i) whose domain includes to blend S to aboundary portion P_(i) of each S_(i), when interpreting S as a functionfrom PS to the common geometric space GS having the geometric objectsS_(i), then S (f_(s) _(i) ⁻¹(P_(i)))⊂P_(i). Additionally, S may be alsocontinuous at each f_(s) _(i) ⁻¹(P_(i)).

Note that Formula (I) above is representative of various formulas forgenerating a blended surface (more generally, geometric object) S, otherembodiments of such formulas are provided in the Detailed Descriptionhereinbelow. Further note that such formulas may be generalized whereinthe parameter space coordinates (u,v) of Formula (I) above can bereplaced by representations of other parameter space coordinates such astriples (u,v,w) or merely a single coordinate u. Moreover, the blendingfunctions B₁ and B₂ may also be defined for such other parameter spaces.Additionally, it is worthwhile to note that such blending functions B₁and B₂ may be considered as weights of a weighted sum of points selectedfrom the surfaces (more generally, geometric objects) S₁ and S₂.Further, this weighted sum notion may also be extended in various ways.For example, referring to the more general embodiment wherein aplurality of geometric objects S_(i), i=1, . . . , N are provided, acorresponding weight/blending function B_(i) may be provided for eachvalue of i so that the following variation of Formula (I) is obtained:

${S(q)} = {\sum\limits_{i = 1}^{N}{{B_{i}(q)}{S_{i}(q)}}}$for points q in a common parameter space for the S_(i).

In another aspect of the present invention, it is within the inventionscope to also generate blended geometric objects S, wherein at leastsome of the S_(i) geometric objects used to generate S are of a higherdimension than 2. For example, if S₁ and S₂ are parameterized solids,then S may be generated as a solid blended from S₁ and S₂ using anothervariation of Formula (I), as one skilled in the art will understand.Thus, S may extend between S₁ and S₂ so that a surface P₁ of S₁ and asurface P₁ of S₂ are also boundaries of S, and S is represented as aweighted sum of points of S₁ and S₂ similar to Formula (I).

In yet another aspect of the present invention, one or more of theparametric geometric objects S_(i) of Formula (I) (or variations thereofsuch as Formula (2), (4), (5), (5.02), (5.03), etc. provided in theDetailed Description hereinbelow) may have one of: a Bezier, NURBS, orsome other multivariant parametric computational representation insteadof, e.g., isocline ribbons as illustratively used in the descriptionherein. Moreover, as one skilled in the art will also understand, it iswithin the scope of the present invention that the underlying geometricobjects that define the S_(i)'s (e.g., for the S_(i) being isoclineribbons, such underlying geometry being markers, profiles, isoclinehandles and profile handles) may be different for a differentcomputational representation. For example, in a Bezier or NURBSrepresentation of an S_(i) “control points” and/or geometric entitiesderived therefrom, may be used to change a shape of the S_(i) andtherefore change the shape of the resulting geometric object S derivedtherefrom.

In another aspect of the present invention, when a blended surface S isgenerated from one or more isocline ribbons S₁, . . . , S_(N), N≧2, thesurface S can be deformed by changing geometric characteristics of theisocline ribbons S_(i). In particular, by changing the shape of one ofthe isocline ribbon boundaries for some S_(i), the points S_(i)(u,v)change and accordingly, the blended surface S changes since it is aweighted sum of such points. In particular, rates of change of geometriccharacteristics of S (such as curvature, tangent vectors, and/or tangentplanes) may be determined by the shape of the isocline ribbons S_(i).More particularly, assuming a substantially linear parameterizationalong each isocline picket, the greater the relative magnitude of suchpickets for a particular isocline ribbon, the more the shape of S willbe skewed in the direction(s) of such pickets. Moreover, as thedirection of such pickets changes, the curvature of S changes. That is,since the weighted sums, such as Formula (I), cause S to always heel tothe surfaces S_(i), the shape of S will change so that S heels to theisocline ribbon(s) S_(i) having pickets whose directions have changed.Thus, the shape of the blended surface S may be changed by any userinteraction technique that: (a) changes one or more geometriccharacteristics of one or more of the S_(i), wherein such changes mayinclude: changing a shape of S_(i)(wherein shape denotes a plurality ofgeometric characteristics such as continuity, differentiability,curvature, and higher order continuity), (b) changes a parameterizationof an S_(i).

Also note that such user interaction techniques for deforming a blendedsurface may also be used with higher dimensional geometric objects. Forexample, if the S_(i) geometric objects are solids rather than surfaces,then a resulting blended solid, S may be deformed by changing a shape ofone or more of the solids S_(i) used in determining S.

It is also within the scope of the present invention that the geometricobjects S_(i) used to generate a blended geometric object S may be suchthat the S_(i)'s can be modified indirectly via other geometric objectsfrom which the S_(i)'s may be themselves generated. For example, if S isa surface blended from isocline ribbons S₁ and S₂ (having correspondingprofiles P₁ and P₂, respectively), and the ribbon S₁ is interpolatedfrom the profile handle, the isocline handle, and the ribbon tangent atthe end points of P₁, then the present invention provides userinteraction techniques for modifying such handles and/or ribbon tangentsfor thereby modifying the blended surface S. Moreover, in one userinterface technique, only the handles may be displayed, wherein suchhandles are displayed as connected to the blended surface S. Thus, bychanging such handles, the blended surface changes. Note that such userinteraction techniques may be responsive in real time to user changes tosuch handles and/or ribbon tangents. Thus, a user's design intent may beimmediately displayed while the user is inputting such changes.Accordingly, using the present invention, user interactions in thedesign process may become closer to the techniques in used inconstructing actual geometric models rather than prior art CAD userinteraction techniques.

It is another aspect of the present invention that various geometricconstraint criteria are capable of being applied to geometric objectsgenerated according to the present invention. In particular, featuresand/or subgeometry of a geometric object O₀ are capable of beingconstrained to lie within another geometric object, O₁, so that as O₁ isdeformed, the features and/or subgeometry of O₀ deform correspondingly,and thereby cause O₀ to deform accordingly. For example, the presentinvention allows an object space point p to be defined (i.e.,parameterized) so that it must remain in/on a given geometric object O₁,where O₁ may be a curve, surface, volume or solid. Thus, as O₁ isdeformed, O₀ also deforms. Moreover, instead of a point p, othergeometric subobjects may also be similarly constrained, such as curves,surfaces or solids. Additionally, features of a geometric object O₀ suchas control points, handles (of various types, e.g., profile andisocline), normals, twist vectors, etc. may also be similarlyconstrained by the present invention so that as O₁ is deformed, O₀ iscaused to also deform. For instance, using the geometric objectinterpolation techniques provided by the present invention, e.g.,Formula (I) and variations thereof, the geometric object O₀ can beefficiently regenerated (e.g., reinterpolated) substantially inreal-time when constrained features and/or subgeometries of O₀ arecorrespondingly deformed with a deformation of O₁. More particularly,this aspect of the present invention provides for the combining ofvarious geometric objects hierarchically so that geometric deformationcontrol of a parent object causes corresponding geometric changes independent child geometric objects. For example, when a surface patchrepresents fine scale detail of a larger surface, it may be advantageousto attach the fine detail surface patch to the larger surface to therebygive a user automatic control over the shape of the fine detail surfacepatch by controlling the shape of the larger surface. Moreover, similarhierarchical control can be provided with other geometric objects oftypes such as curves, points and three-dimensional deformation spaces.

Note that such hierarchical control may be also used with a persistentdeformation space wherein it is desirable for a geometric object in thisspace to be repeatedly deformed and restored to its originalnon-deformed state. Note that this is difficult to do in real-time byrepeatedly applying a one-time deformation. Accordingly, by utilizingsuch hierarchical control of the present invention, a geometric objectembedded in such a three-dimensional deformation space and/or thecontrol structures of the geometric object embedded therein provides forthe deformation of the geometric object when the three-dimensionaldeformation space is deformed. Further, if one or more such deformationspaces are, in turn, made dependent upon a simpler geometry such as asurface or curve, then substantial control over the shape of thegeometric object, however complex, can be provided by manipulating theshape of the simpler geometry.

Other features and benefits of the present invention are provided in theDetailed Description and the drawings provided herewith.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a surface 62 generated according to the present invention,wherein the surface interpolates between the surfaces 30 and 34, andalso passes through the curves 54, 58 and 60 at predetermined directionsaccording to the isocline ribbons 61 and 63;

FIG. 2 shows a further modification of the surfaces of FIG. 1, whereinthe surface 30 has a circular disk 66 blended thereto according to amethod of the present invention;

FIG. 3 shows a blended surface 62 a generated according to the presentinvention between the surfaces 30 and 34, wherein the surface 62 apasses through the curves 54 and 58 and wherein the blending isperformed according to a novel surface generation formula providedherein (Formula (I));

FIG. 4 illustrates a correspondence between geometric entities inparameter space and geometric entities in object space, wherein lines 78a and 78 b of parameter space have object space images of curve 54 and58, respectively, and additionally, parameter space line 86 has as anobject space image curve 80;

FIG. 5 provides a graphical representation of two blending functions, B₁and B₂, utilized in some embodiments of the present invention;

FIGS. 6A–6D show graphs of additional blending functions that may beused with the present invention;

FIG. 7 provides a further illustration of the correspondences betweengeometric entities in parameter space and object space;

FIG. 8 shows an elliptic region 100 that is blended into a cylinder 108according to the present invention, wherein the closed curve 110delimits the elliptic region from the deformed portion of the cylinder108 that blends to the closed curve;

FIG. 9 shows a simple boss 112 created on a cylinder 116 according tothe method of the present invention;

FIG. 10 shows a composite curve 120 (as defined hereinbelow) thatincludes two crossing subcurves 124 and 128;

FIG. 11 shows a surface 130 from which a label surface 134 is trimmed;

FIG. 12 illustrates one computational technique for determining adistance-like measurement from a point p that is interior to the polygonhaving vertices v₁, v₂, v₃, v₄, and v₅, wherein the distance-likemeasurement to each side of this polygon is determined using acorresponding apex 150 provided by a stellating process;

FIG. 13 shows two boundary curves 156 a and 156 b in parameter space(i.e., the unit square), wherein area patches 168 therebetween arecapable of being themselves parameterized by coordinates (s,t) where svaries linearly with the distance between two corresponding points on apair of opposing subcurves 160 a and 160 b, and t determines acorresponding point on each of the subcurves of the pair 160 a and 160b;

FIG. 14 illustrates a region that has sides and ribbons defined by threesurfaces S₁, S₂ and S₃, wherein the present invention is able to providea surface patch for the region 300 using Formula (5) providedhereinbelow;

FIGS. 15 and 16 illustrate both general computational geometry concepts,as well as novel concepts that are fundamental to the present invention.Note that these figures are used to illustrate the terms defined in theDefinitions Section hereinabove;

FIG. 17 shows a block diagram of the typical flow of design constructionoperations performed by a user of the present invention in designing ageometric object;

FIG. 18 shows three profile curves 404, 408 and 412 meeting at a profilemarker 420, wherein the surfaces 416 and 418 join smoothly at the marker420 due to the isocline handles (for the marker 420) being in a commonplane 460;

FIG. 19 shows profile curves x and y that define a surface 480 whichforms a fillet between surfaces 484 and 486. Typically, profiles x and yare defined using distances 488 and 490 from the intersection curve 482of surfaces 484 and 486;

FIG. 20 illustrates one embodiment for computing a blended surface fromisocline ribbons 508 and 516 according to the present invention;

FIGS. 21A–21C illustrate a procedure for creating a hole 600 accordingto the present invention;

FIG. 22 shows a blended surface 710 according to the present invention,wherein the blended surface extends between a degenerate profile (point)714 and the circular end 718 of a cylinder 722;

FIG. 23 shows a blended surface 750 according to the present inventionthat extends between the degenerate profile (point) 754 and the planardisk 758 having a circular curve 760 therein;

FIG. 24 illustrates the results of a blending technique of the presentinvention for blending a surface between semi-circular ribbons 784 a and784 b, wherein the resulting surface 786 is blended between these tworibbons.

FIG. 25 shows a blended surface 808 according to the present inventionwhose points p(u,v) are determined using a “forward algorithm”, whereinpoints in parameter space 158 are themselves parameterized according topoints in an additional parameter space 828, and wherein the points 830of the additional parameter space are used to efficiently determine thedistance-like measurements to the pre-images (in parameter space 158) ofthe profiles 812 and 816 (in object space);

FIG. 26 is a flowchart showing the steps for computing an interpolatingcurve according to the present invention using a one-dimensionalembodiment of the computational techniques novel to the presentinvention;

FIG. 27 shows a flowchart of the steps performed when constructing anapproximation to an isocline boundary of an isocline ribbon, wherein theboundary is opposite the profile for the isocline ribbon;

FIGS. 28A and 28B show a flowchart for a program that constructs a moreprecise isocline ribbon boundary than the approximation resulting fromFIG. 27;

FIGS. 29A–29C illustrate a flowchart for modifying one or moresubsurfaces S_(i) of a composite surface S₀ by changing a geometriccharacteristic of an isocline handle and/or a ribbon tangent for amarker on one or more profile curves defining the boundaries for thesubsurfaces S_(i);

FIGS. 30A and 30B provide a flowchart of a program invoked by theflowchart of FIGS. 29 for deforming subsurfaces S_(i) in real time as auser modifies an isocline handle and/or ribbon tangent;

FIG. 31 is a flowchart of the high level steps performed by a userinteracting with an embodiment of the present invention for changing theshape of a surface;

FIG. 32 pictorially illustrates examples of values for parameters usedin the flowchart of FIG. 26 for computing an interpolating curve C(u);

FIG. 33 shows four profile curves P₁₁, P₁₂, P₂₁ and P₂₂ wherein it isdesired to generate a surface, bounded by these profiles and wherein thesurface is defined by these four profiles (and their associated isoclineribbons);

FIGS. 34 and 35 illustrate the intermediary surfaces generated duringthe performance of one method for creating a 4-sided patch (FIG. 36)from two 2-sided blends using the four profile curves of FIG. 33. Thatis, a blended surface S₁ (FIG. 34) is generated using the isoclineribbons R₁₁, and R₁₂ (for the profiles P₁₁ and P₁₂, respectively), and ablended surface S₂ (FIG. 35) is generated using the isocline ribbons R₂₁and R₂₂ (for the profiles P₁₂ and P₂₂, respectively);

FIG. 36 shows a resulting blended surface S derived from S₁ (shown inFIG. 34), and S₂ (shown in FIG. 35), wherein S is generated according toFormula (II) provided in Section 5 of the detailed descriptionhereinbelow;

FIG. 37 shows the geometric objects used in an embodiment of the presentinvention for generating a surface S from two surfaces S_(L) and S_(R).In particular, this figure introduces the notational conventionssubsequently used in FIGS. 38 through 41;

FIG. 38 illustrates one embodiment of the present invention forgenerating a four-sided patch;

FIG. 39 illustrates an alternative embodiment of the present inventionfor generating the four-sided patch also generated in FIG. 38;

FIG. 40 shows the notational correspondences between the geometricobjects of FIG. 38 and those of FIG. 39;

FIG. 41 shows a possible geometric configuration of FIG. 38, wherein theprofiles P₃ and P₄ of FIG. 38 are degenerate;

FIGS. 42A and 42B illustrate the movement of a marker 2002 that isconstrained to reside on the profiles curves 2003 and 2004;

FIG. 43 illustrates constraints on composed profile curves and theircorresponding isocline ribbons for providing tangent plane continuitybetween two blended surfaces S₁ and S₂;

FIG. 44 shows a profile P, associated isocline ribbons RL and RR, andvarious handles used in describing the conditions for achieving GIcontinuity on P.

DETAILED DESCRIPTION 1. Introduction

FIG. 1 illustrates the use of an embodiment of the present invention fordesigning a surface 62 that interpolates any two parametric surfacessuch as between the half cylinder surface 30 and the surface 34. Thatis, the surface 62 is generated via a novel surface interpolationprocess, wherein constraints on surface 62 shape are provided by thefeature curves 54, 58 and 60, and their associated novel controlgeometry (e.g., isocline ribbons). In particular, the followingconstraints are satisfied by the surface 62:

-   -   (a) one or more geometric characteristics of the surface 30        along the feature curve 54 are imposed on the surface 62,    -   (b) one or more geometric characteristics of the surface 34        along the feature curve 58 are imposed on the surface 62, and    -   (c) the surface 62 interpolates through the feature curve 60,        wherein the surface 62 tangents along the extent of curve 60 are        derived from (e.g., identical to) the isocline ribbons 61 and        63.        Thus, using the present invention, a designer can design a        surface specified in terms of: (a) a relatively small number of        carefully constructed and positioned feature curves, and (b) the        desired slope(s) of the surface along the extent of these        feature curves (via isocline ribbons). Moreover, using the        present invention, such a designed surface not only interpolates        fairly between the feature curves but also may obey other        imposed constraints such as convexity, concavity, and/or        predetermined curvature ranges.

Additionally, the present invention can be used to blend a surfaceregion into an object being designed. For example, FIG. 2 illustratesthe blending of a circular disk 66 into the cylindrical surface 30.Moreover, the present invention can also be used to construct bosses,dimples, logos, and embossing as well as to recursively design surfacesas one skilled in the art will come to appreciate from the disclosureherein.

At least one embodiment of the present invention differs fromtraditional approaches to computer-aided design (CAD) in that with thepresent invention, a desired geometric object (e.g., a surface) that maybe created as a plurality of geometrically and computationally unrelatedpatches (e.g., three-, four-, five-sided bounded surfaces), which may besubsequently pieced together in a way that satisfies certain constraintsat the boundaries between the patches. Thus, the desired geometricobject can be designed by piecing together the plurality of unrelatedgeometric sub-objects (subsurfaces) in a manner that interpolates,blends and/or trims these sub-objects so that, across the boundariesand/or regions therebetween, constraints such as continuity,differentiability, and/or curvature are satisfied. This is fundamentallydifferent from the traditional approaches to CAD in that only four-sidedNURBS, Bezier, Hermite, Coons, Gordon or Booleans of implicit surfacesare patched together in prior art systems.

2. Blending between Geometric Objects

A fundamental geometric object design technique of the present inventionis the blending between two parametric geometric objects such assurfaces and, more particularly, the manner in which such blending isperformed. As defined in the Definitions Section above, a “parametricgeometric object” (e.g. a surface) may be defined as a result of amapping from a (simple) coordinatized geometric object (parameter space)such as a bounded plane to another (typically, more complex) geometricobject (object space). When the parameter space is a bounded plane, twocoordinates or parameters (denoted u and v) may by way of example beused to uniquely identify each point in the parameter space. When theobject space is three-dimensional, for every (u,v) point in the boundedplane parameter space, a function may associate a point (x,y,z) in theobject space.

By convention, a planar parameter space is usually assumed to be theunit square, which means that both u and v vary between 0 and 1,although it is within the scope of the present invention to utilizeother parameter space geometries and coordinate ranges.

In one embodiment of the present invention, in order to blend betweentwo parametric surfaces S₁(u,v) and S₂(u,v) each having the unit squareas their parameter space, each surface S₁ and S₂ has associatedtherewith a respective blending function B₁(u,v) and B₂(u,v), whereineach of the blending functions has, for example, (0,1) as its range (aswell as satisfying other properties given hereinbelow). Consequently, anew surface may be defined by the following formula:S(u,v)=S ₁(u,v)·B ₁(u,v)+S ₂(u,v)·B ₂(u,v)  (1)

Note that the blending functions B₁ and B₂ are typically chosen so thatthe resulting blended surface S is the same as S₁ on a boundary with S₁,and the same as S₂ on a boundary with S₂. This is achieved by devisingB₁ and B₂ so that B₁=1 and B₂=0 on the boundary with S₁ while havingB₂=1 and B₁=0 on the boundary with S₂. In FIG. 3, for example, if S₁ isthe surface 30 and S₂ the strip 34, and one boundary is the verticalline 54 of the surface 30 and the other boundary is the curve 58 on thestrip 34, then the surface 62 a is S, which runs between these twoboundaries and is tangent to S₁ and S₂ at the boundaries.

2.1. The Blending Functions

Blending functions may be provided for blending between geometricobjects of various types. For example, blending functions for blendingbetween two volume filling geometric objects can be provided. However,to simplify (and clearly illustrate) 20 the novel blending process andthe associated blending functions of the present invention, thediscussion here is initially limited to blending between two curves, orblending between two surfaces. Accordingly, for two surfaces S₁ and S₂to be blended together, the blending functions B₁(u,v) and B₂(u,v),respectively, are appropriately set to either 0 or 1 on the boundariesof a blended surface generated by the present invention.

Referring to FIG. 4, wherein it is assumed that the boundaries 78 a, 78b in parameter space correspond to the profiles 54 and 58 in objectspace, for any curve 80 on the blended surface such as surface 62 a,there is a related pre-image (e.g., line 86) defined in parameter spaceas indicated. Note that for simplicity the boundaries 78 a and 78 b, andthe pre-image of curve 80 are straight, but they need not be so.

Assuming (again for simplicity) the blending functions B₁(u,v) andB₂(u,v) have their domains in the unit square (as their parameterspace), for any point (u, v) in this parameter space it is important todetermine some measure of how “close” the point (u,v) is to the boundarycurves (e.g., boundary curves 78 a and 78 b) and, more generally, to thepre-images of profile curves. Such closeness or distance-likemeasurements may be used in specifying the blending functions and/ortheir resulting values. Note that there are many ways to compute such acloseness or distance-like measurement in parameter space. For instance,if a boundary 78 (or profile pre-image) is a straight line, then such aparametric distance to a (u,v) point is easily calculated as the lengthof a perpendicular line segment to the boundary line through the point.Additional techniques for computing parametric distances are describedhereinbelow (e.g., Sections 2.3 and 2.4).

Assuming parameter space is still the 2-dimensional space of (u,v)points, a blending function {tilde over (B)}_(i) (wherein 1≦i≦N for somefixed N number of boundary curves) can be computed a function of aunivariate distance-like function {tilde over (B)}_(i) (D_(i)), whereD_(i) is in turn a function of (u,v) so that {tilde over(B)}_(i)(D_(i))={tilde over (B)}_(i()D_(i) (u,v))=B_(i) (u,v), whereinD_(i)(u,v) is a distance-like function to the pre-image C_(i) ⁻¹ of aboundary curve C_(i) (in object space) of a surface S. Note that suchdistance-like functions must satisfy the condition that as (u,v) getsarbitrarily close to the i^(th) boundary curve pre-image C_(i) ⁻¹ (suchas measured in conventional Euclidian distance), then D_(i)(u,v) getsarbitrarily close to zero. Examples of such blending functions {tildeover (B)}_(i) and distance-like functions D_(i) are providedhereinbelow.

Since many of the most useful blending functions B_(i) are of the form{tilde over (B)}_(i) (D_(i)), unless additional specificity is required,B_(i) will be used hereinbelow to denote both: (a) the blending functionB_(i)(u,v) initially discussed above, and (b) the blending function{tilde over (B)}_(i) (D_(i)) for some distance-like function D_(i). If,however, a clear distinction is required between the blending functionsof (a) and (b), the domain of the blending function can be used toindicate which blending function is indicated. As an aside, note thatFormula (I) applies equally well for the blending functions {tilde over(B)}_(i) (D_(i)), i=1, 2; that is,S(u,v)=S ₁(u,v)·{tilde over (B)} ₁(D ₁)+S ₂(u,v)·{tilde over (B)} ₂(D₂).  (2)

If a point (u,v) is close to the i^(th) pre-image boundary i=1,2, then{tilde over (B)}_(i) (D_(i)) is expected to be small and the point ismapped (into object space) close to the i^(th) boundary.

A good collection of blending functions B_(i) not only allows themapping, S, of a blended surface to be coincident with the desiredperimeter (profile) curves, but will do so in a manner so that theresulting blended surface between two or more such perimeter curves of,e.g. for example, different initial surfaces will preserve suchcharacteristics as the continuity of curvature with these initial twosurfaces. That is, the blended surface “heels” to each of the initialsurfaces. Also, it is preferred that the blending functions B_(i) allowthe new surface to be fair. FIG. 5 shows a graph of a pair of desirableblending functions for B_(i), i=1,2.

For profile curves P₁, P₂ of two surfaces S₁ and S₂, wherein a blendedsurface is desired between P₁ and P₂, assume that the profiles P₁ and P₂have parametric pre-images that correspond, respectively, to u=0, u=1 ofthe unit square {(u,v)|0≦u≦1 and 0≦v≦1}, then some useful properties forblending functions B₁ and B₂ are:

-   -   (1.1) B₁=1 at u=0 and B₁=0 at u=1. B₂=0 at u=0 and B₂=1 at u=1.    -   (1.2) The derivatives B₁′ and B₂′ equal 0 wherever D_(i)(u,v)=0        and D_(i)(u,v)=1, i=1,2. This enforces smooth (tangent        continuous) transitions between the blended surface S and the        initial surfaces S₁ and S₂. If higher order derivatives are also        zero, then higher order continuity between surfaces can be        realized, usually improving its fairness.    -   (1.3) B₁+B₂=1 for all points (u,v). This is called a “partition        of unity,” and it provides for the generation of a convex        combination of the surfaces S₁ and S₂ to which a new blended        surface abuts. Note that this tends to keep the new blended        surface from drifting too far from the initial surfaces S₁ and        S₂.

There are numerous embodiments for defining blending functions. Oneuseful embodiment is:

$\begin{matrix}{{B_{1}\left( D_{1} \right)} = {{{\cos^{2}\left( {D_{1}\frac{\pi}{4}} \right)}\mspace{31mu}{and}{\mspace{14mu}\mspace{14mu}}{B_{2}\left( D_{2} \right)}} = {\sin^{2}\left( {D_{2}\frac{\pi}{4}} \right)}}} & (3)\end{matrix}$which gives arbitrarily high order continuity of the blending functions,which is needed to achieve the same high order continuity between theinitial blended surfaces. Another alternative is to choose polynomialfunctions with the above properties (1.1) through (1.3). For example, aquintic polynomial can be chosen with zero second derivative at D=0 andD=1, thereby providing beneficial curvature characteristics (see Section4.4).

In addition to the blending functions described hereinabove, thefollowing are examples of additional blending functions:

-   -   B₁(x) and B₂(x) are polynomials satisfying the following        constraints:        -   B₁(0)=1, B′₁(0)=0, B(1)=0, and B′(1)=0        -   B₂(x)=1−B₁(x)        -   Note that additional constraints regarding high order            derivatives (e.g., equal to 0 at x=0 and/or 1) may also be            imposed. For example, if B″_(i)(0)=B″_(i)(1)=0, i=1,2, then            C² continuity is attained with the objects from which            interpolating and/or blending is performed.        -   An example of polynomial blending functions satisfying these            constraints is:        -   B₁(x)=(1−x)²+5x(1−x)⁴+10x²(1−x)³        -   B₂(x)=1−B₁(x).    -   Note that B₁(x) may be derived as a Bezier curve with six        control points, P₁, . . . P₆, as shown in FIG. 6D. Moreover,        note that since        -   B′₁(x)=−30x²·(1−x)² and        -   B″₂(x)=60x·(1−x)²−60x²·(1−x),    -   that        -   B′₁(1)=0, B″(0)=0, B″₁(1)=0, B′₂(0)=0, B′₂(1)=0, B″₂(1)=0,            B′₁(0)=0 and        -   B″₂(0)=0.    -   (c) Any composition of blending functions as described        hereinabove with a bijective (e.g., one-to-one and onto)        parameterization function P:[0,1]→[0,1] may be composed with a        blending function to obtain another blending function. As a        specific example, let P(x)=2c(x−x²)+x², where c is a constant        “skew” factor, then a new blending function may be construed as        B(P(x)). Thus, when c=½, P(x)=x. Moreover, as c varies, the        inflection point of the graph of P(x) moves as shown in FIGS.        6A–6C. Note that the blending function of FIG. 6B (wherein c>½)        will cause the blending curve (and/or surface or other geometric        object) to retain the geometric characteristics of the object        used for blending that corresponds to the x=0 axis in the graph        of FIG. 6B for a larger part of the surface.

To reduce the real-time design computational overhead incurred forevaluating blending functions, the values for the blending functionsmay, in one embodiment, be tabulated prior to a design session at asufficiently high resolution and stored in memory in a manner thatallows efficient indexed access to a closest approximation to the actualblending function value.

2.2. Extending Blending to N-Sided Regions

In one embodiment of the present invention, a novel general form forblending over a region that is bounded by each edge e_(i) of aparametric surface S_(i) (i=1,2, . . . , N and N≧2) is the followingweighted sum of points S_(i)(u_(i)(p),v_(i)(p)):

$\begin{matrix}{{S(p)} = {\sum\limits_{i = 1}^{N}{\left\lbrack {\prod\limits_{\underset{j \neq i}{j = 1}}^{N}\;{B_{j}\left( {D_{j}(p)} \right)}} \right\rbrack{S_{i}\left( {{u_{i}(p)},{v_{i}(p)}} \right)}}}} & (4)\end{matrix}$where:

-   -   (a) p is a variable denoting points in a common parameter space        for the surfaces S_(i);    -   (b) D_(j)(p) is a distance-like measurement to the pre-image of        the i^(th) edge e_(i) in the common parameter space;    -   (c) B_(j) is a blending function which is zero when D_(j) is        zero and monotonically increases as D_(j) increases; and    -   (d) u_(i) and v_(i) are parameterization functions that        transform p from the common parameter space to the (any)        intermediate parameter space for S_(i).        Note that by dividing by the sum of the products of the blending        functions, B_(j),

$\left( {{e.\mspace{11mu} g.},{\sum\limits_{i = 1}^{N}\left\lbrack {\prod\limits_{\underset{j \neq i}{j = 1}}^{N}\; B_{j}} \right\rbrack}} \right)$the formula (4) can be normalized with respect to the blendingfunctions. Further note that Formula (4) resembles Formula (I) when N=2,and is in fact an extension thereof. That is, for N=2, B₁ of Formula (4)has the functional behavior of B₂ in Formula (I), and B₂ of Formula (4)has the functional behavior of B₁ in Formula (I). That is, there is asubscripting notational change between the two formulas.

As an example of Formula (4), consider the three-sided region 300 shown(in object space) in FIG. 14. Applying Formula (4) to thereby generate asurface, S, for region 300, the following equation is obtained:S(p)=B ₂(v)B ₃(w)S ₁(u)+B ₁(u)B ₃(w)S ₂(V)+B ₁(u)B ₂(v)S ₃(w)  (5)where u, v and w as parameterization functions are the barycentriccoordinates of p as one skilled in the art will understand.

An alternative method to define a blended surface over N-sided (N≧4)regions is provided by first applying the two-sided approach based onFormula(1) using R₁₁ and R₁₂ of FIG. 34 as S₁ and S₂, respectively inFormula (1) to thereby generate S₁ of FIG. 34. Additionally, Formula (1)is applied to the surfaces of FIG. 35, wherein S₁ and S₂ of Formula (1)are replaced by R₂₁ and R₂₂ respectively, to thereby generate S₂ of FIG.35. The two resulting surfaces S₁ and S₂ of FIGS. 34 and 35 respectivelyare, in turn, blended using Formula (2) wherein blending functions B₁and B₂ are as described hereinabove, and the corresponding D_(i) aredescribed hereinbelow For example, given that each of the ribbons R₁₁,R₁₂, R₂₁, and R₂₂, have a common pre-image, the D_(i) used in Formula(2) to compute distance-like measurements to the pre-images of the pairof edges P₁₁, P₁₂, P₂₁, and P₂₂ (FIGS. 34 and 35) may be:

(a) For a point P₁ of the (common) pre-image for S₁ of FIG. 34,D₁(P₁)=min (D(P₁,P₁₁₎), D(P₁,P₁₂)) wherein D is the Euclidean distancebetween P₁ and the corresponding profile P_(1i), and

(b) For a point P₂ of the (common) pre-image for S₂ of FIG. 35,D₂(P₂)=min (D(P₂,P₂₁₎), D(P₂,P₂₂)). Accordingly, the two surfaces S₁ andS₂ can be blended together using Formula (2) to obtain surface S of FIG.36.

In another embodiment that is particularly useful for generating afour-sided blended patch, assume the following restricted but versatilescheme for defining profiles and ribbons:

(a) All handles are piecewise linear segments; and

(b) All blending is done with the functions B₁(x) and B₂(x) of Formulas(3.1).

Moreover, referring first to FIG. 37 in describing the present patchgeneration technique, the following labeling scheme is used. For theprofile, P:

m_(L), m_(R): the left and right hand markers, respectively, of theprofile, P;

h_(L), h_(R): the left and right hand profile handles, respectively, ofthe profile, P;

s_(L), s_(R): the left and right hand isocline handles, respectively, ofthe profile, P;

b_(L), b_(R): the left and right hand ribbon tangents at the respectiveleft and right end points of isocline boundary R (these ribbon tangentsalso being denoted as “boundary handles”).

Using the notation of FIG. 37, surfaces S_(L) and S_(R) may be defined,wherein S_(L) is bounded by the line segments corresponding to: s_(L),h_(L), b_(L), and d_(L)=(s_(L)+b_(L))−h_(L), and S_(R) is bounded by theline segments corresponding to: s_(R), h_(R), b_(R), andd_(R)=(s_(R)+b_(R))−h_(R). In particular, S_(L) and S_(R)are known inthe art as “twisted flats,” and accordingly, S_(L) is denoted as theleft twisted flat, and S_(R) is denoted as the right twisted flat.Moreover, these surfaces may be evaluated using the following formulas(5.01a) and (5.01b):

$\begin{matrix}{{S_{L}\left( {u,v} \right)} = {\left( {{1 - v},v} \right)\begin{pmatrix}m_{L} & h_{L} \\s_{L} & b_{L}\end{pmatrix}\begin{pmatrix}{1 - u} \\u\end{pmatrix}}} & \left( {5.01a} \right)\end{matrix}$wherein the parameters u and v increase in transverse directions asillustrated by the u-direction arrow and the v-direction arrow (FIG.37).

$\begin{matrix}{{S_{R}\left( {u,v} \right)} = {\left( {{1 - v},v} \right)\begin{pmatrix}h_{R} & m_{R} \\b_{R} & s_{R}\end{pmatrix}\begin{pmatrix}{1 - u} \\u\end{pmatrix}}} & \left( {5.01b} \right)\end{matrix}$wherein the parameters u and v also increase in transverse direction,with the u-direction being the reverse direction of the u-directionarrow of FIG. 37.

Accordingly, the isocline ribbon surface S (FIG. 37) can now be definedas follows:S(u,v)=B ₂(u)S _(L)(u,v)+B ₁(u)S _(R)(u,v)  (5.02)where conveniently, the u parameter is also the distance measure neededfor B₁ and B₂ of Formulas (3.1). Thus, when v=0, S(u,0) is the profile;i.e., a blend between the control handles (h_(L)−m_(L)) and(h_(R)−m_(R)). Additionally, note that when v=1, S(u,1) is the ribbonboundary R derived as a blend of vectors (b_(L)−s_(L)) and(b_(R)−s_(R)). Also note that if b_(L) and b_(R) are translates of h_(L)and h_(R), respectively, along s_(L)−m_(L) and s_(R)−m_(R),respectively, then R is a translation of P, and such similarities maysimplify the data storage requirements of the present invention.

For a plurality of isocline ribbons S₁, S₂, . . . , S_(N), wherein eachS_(i) is generated by Formula (5.02), such ribbons may now be used inthe more general N-sided surface form below, which is a variation ofFormula (4).

$\begin{matrix}{{S\left( {s,t} \right)} = \frac{\sum\limits_{i = 1}^{N}{\left( {\prod\limits_{\underset{j = 1}{j \neq i}}^{N}{B_{j}\left( {D_{j}\left( {s,t} \right)} \right)}} \right){S_{i}\left( {{u_{i}\left( {s,t} \right)},{v_{i}\left( {s,t} \right)}} \right)}}}{\sum\limits_{i = 1}^{N}\left( {\prod\limits_{\underset{j = 1}{j \neq i}}^{N}{B_{j}\left( {D_{j}\left( {s,t} \right)} \right)}} \right)}} & (5.03)\end{matrix}$Note that D_(j)(s,t), u_(i)(s,t) and v_(i)(s,t) must be defined for thisformula, i.e., the distance measure and the mappings from the generalN-side patch parameter space (in s and t) to the parameter space of theribbons S_(i) (in u and v).

For specific cases where N=2, 3, 4 and N≧5 using the blended ribbonsS_(i), notice first that Formula (5.02) for the ribbon is a special caseof Formula (5.03). For example, in Formula (5.02) the denominator is 1,the distance measure is just the u-parameter, and u and v correspondexactly to s and t. The formula for a two-sided surface is similar,except that the base surfaces are ribbons derived according to Formula(5.02) (denoted herein also a “twisted ribbons”); thus,S(u,v)=B ₂(v)S ₁(u,v)+B ₁(v)S ₂(u,v)  (5.04)which is similar to Formula (5.02),wherein the parameter u measuresdistance. It varies along the direction of the profile curve. However,in Formula (5.04), the parameter v measures distance.

Referring to FIG. 38, wherein the isocline ribbons S₁ and S₂ areparameterized as indicated by the u and v direction arrows on each ofthese ribbons, these ribbons may be used to generate a four-sided patch.The two profiles P₁ and P₂ that vary in u are blended using the twistedribbons S₁ and S₂. The other two sides P₃ and P₄ are blended profilesderived from the isocline handles; that is, P₃ is a blend (e.g., usingFormula (1)) of h_(R) ¹ and h_(L) ¹, wherein h_(R) ¹ is S₁ and h_(L) ¹is S₂ in Formula (1), and similarly, P₄ is a blend of h_(R) ² and h_(L)².

Note that the blended surface, S, of FIG. 38 has tensor product form.This can be shown by decomposing Formula (5.04) into a tensor form,wherein each of the ribbons S₁ and S₂ is derived from the Formula(5.01a) and (5.01b). That is, S₁ is a blend of S_(L) ¹ and S_(R) ¹ (FIG.38) and S₂ is a blend of S_(R) ¹ and S_(R) ². Accordingly, thedecomposition is as follows:

$\begin{matrix}\begin{matrix}{{S\left( {u,v} \right)} = {\left( {{B_{2}(v)}{B_{1}(v)}} \right)\begin{pmatrix}{S_{1}\left( {u,v} \right)} \\{S_{2}\left( {u,v} \right)}\end{pmatrix}}} \\{= {\left( {{B_{2}(v)}{B_{1}(v)}} \right)\begin{pmatrix}{S_{L}^{1}\left( {u,v} \right)} & {S_{R}^{1}\left( {u,v} \right)} \\{S_{L}^{2}\left( {u,v} \right)} & {S_{R}^{2}\left( {u,v} \right)}\end{pmatrix}\begin{pmatrix}\left. {B_{2}u} \right) \\{B_{1}(u)}\end{pmatrix}}} \\{= {\left( {{{B_{2}(v)}S_{L}^{1}} + {{B_{1}(v)}S_{L}^{2}{B_{2}(v)}S_{R}^{1}} + {{B_{1}(v)}S_{R}^{2}}} \right)\begin{pmatrix}\left. {B_{2}u} \right) \\{B_{1}(u)}\end{pmatrix}}}\end{matrix} & (5.05)\end{matrix}$Thus, the last expression above shows that the same surface S can begenerated by first creating the twisted ribbons in the vparameterization, and then second, blending in u. However, since theroles of u and v are symmetric, the twisted ribbons may be generatedalong the u parameterization and subsequently, the blending may beperformed in v. That is, using the surfaces S_(L) ³ and S_(R) ³, S_(L) ⁴and S_(R) ⁴ of FIG. 39 gives the same surface S as in FIG. 38. Thus, ineither technique for deriving S, the inputs are the same; that is, m_(L)^(i), M_(R) ^(i), h_(L) ^(i), h_(R) ^(i), S_(L) ^(i), S_(R) ^(i), b_(L)^(i), and b_(R) ^(i), where “i” denotes the profile P_(i)(i=1, 2, 3, 4)to which the inputs apply. Note that the correspondences between thevarious inputs is shown in FIG. 40.

So, overall, the two-sided patch of Formula (5.04) provides a veryversatile four-sided patch. Moreover, its evaluation is also efficient.Thus, by expanding the S_(L) ^(i) and S_(R) ^(i) of Formula (5.05) usingFormulas (5.01a) and (5.01b), the following expression may be obtained:

$\begin{matrix}{{\left( {{B_{2}(v)},{B_{1}(v)}} \right)\begin{bmatrix}\left( {{1 - v},v} \right) & \begin{pmatrix}\begin{pmatrix}1 \\L\end{pmatrix} & \begin{pmatrix}1 \\R\end{pmatrix} \\\begin{pmatrix}2 \\L\end{pmatrix} & \begin{pmatrix}2 \\R\end{pmatrix}\end{pmatrix}\end{bmatrix}}\begin{pmatrix}{1 - u} \\u\end{pmatrix}\begin{pmatrix}{B_{2}(u)} \\{B_{1}(u)}\end{pmatrix}} & (5.06)\end{matrix}$where

$\begin{pmatrix}i \\L\end{pmatrix}\mspace{14mu}{and}\mspace{14mu}\begin{pmatrix}i \\R\end{pmatrix}$are the appropriate matrices from Formulas (5.01a) and (5.01b). Notethat when evaluating an instantiation of this expression, the B_(i)should probably be table driven.

The above formulation is mathematically sound, but to use it in ageometrically intuitive fashion still requires judgment on the user'spart. Thus, in certain degenerate cases, some mathematical aids are alsoin order. A common instance is where two of the profiles (e.g., P₁ andP₂) intersect each other, as in FIG. 41. This is a degenerate case sinceprofiles P₃ and P₄ (of FIG. 38) are zero length, and share end markers(i.e., m_(L) ³=m_(L) ⁴ and m_(R) ⁴=m_(L) ⁴).

Note, however, that Formula (5.04) still defines a surface S, but it iseasy to see that the surface may loop at the profile intersections. Toeliminate this looping and still maintain handle-like control at themarkers, the twisted ribbon of Formula (5.04) may be sealed by afunction of u. One function that is 1 at u=½ and 0 at u=0, is:

$\begin{matrix}{{\alpha(u)} = {1 - {4\left( {u - \frac{1}{2}} \right)^{2}}}} & (5.07)\end{matrix}$Thus, Formula (5.01)O is adjusted to be:S(u,v)=B ₂(v)α(u)S ₁(u,v)+B ₁(v)α(u)S ₂(u.v)  (5.08)Such a function (5.08) will likely remove most loops.

The ability to diminish the ribbon at the ends suggests otherapplications. A scaling function such asα₁(u)=1−u ²  (5.09)diminishes the ribbon at the u=1 end, whileα₁(u)=1−(u−1)²  (5.091)diminishes it at the u=0 end. This is an effective way to make atriangular (three-sided) surface, as one skilled in the art willunderstand.

2.2.1 Bosses and Dimples from 2-Edges

The so-called “boss” feature may be obtained from a blending between twoprofile edges. The profiles may be provided as, for example, semicircles780 a and 780 b of FIG. 24 having isocline ribbons 784 a and 784 b,respectively. The ribbons 784 a and 784 b are in distinct parallelplanes. When these ribbons are blended together, a surface 786 (FIG. 24)is obtained which may be considered a boss or a dimple. Note that manyvariations, i.e., domes, rocket tips, mesas, apple tops, etc. may begenerated similarly. Moreover, if the top semicircular ribbon isrotated, the boss can be made to twist. This scheme can be used totransition between tubes, like a joint, as one skilled in the art willunderstand.

Note that in another embodiment, blending may be performed by using aneighborhood about each boundary curve (in object space) as a defaultisocline ribbon from which to blend using Formula (1) or Formula (4).Thus, by defining a value ε>0, and taking a strip and width of eachsurface along the boundary to which the surface is to be blended, thesestrips may be used as isocline ribbons. Accordingly, the surfaceboundaries become profile curves and pre-images thereof may be used inthe Formula (1) or Formula (4).

2.3. Profile Curves

Since the present invention can take a few well-positioned (objectspace) profile curves of various types and generate a correspondingsurface therethrough, as a blended surface according to Formula (1)above, there are two parameter space pre-image curves for each of thesurfaces S₁ and S₂ wherein these curves are boundaries for the blendingfunctions B₁ and B₁; that is, a curve at D_(i)=0 and at D_(i)=1 for eachblending function B_(i). In fact, there may be eight curves, asillustrated in FIG. 7, that may be used to define a blended surface.That is, there may be two curves 78 a and 78 b in the parameter space ofS₁ and two additional curves 78 c and 78 d in the parameter space for S₂(of course, in many cases these two parameter spaces are identified).Additionally, there are the mappings of the curves 78 to the twosurfaces 30 and 34, thereby providing the corresponding image curves 90,54, 58 and 91, these having respective pre-images 78 a, 78 b, 78 c and78 d.

Note that in the case where S₁ and S₂ have identical parameter spaces,profile 78 b is the pre-image of the profile 54. Moreover, if S₂ of 78 d(=78 b) is profile 58, then 78 b is included in the pre-image of each ofS₁, S₂ and blended surface 62.

When the present invention is used for surface design, a user ordesigner may think of designing a blended surface by continuouslypulling or deforming one profile curve of an initial surface to therebycreate a new surface between this initial surface and a profile curve ofanother initial surface.

Note that different types of profile or boundary curves may be used withthe present invention. In some embodiments of the present invention,such a profile curve, C, may typically have a parametric pre-image in aparameter space, i.e. C⁻¹(s)=(u(s),v(s)) where s is a parameterizationof the pre-image (e.g., 0≦s≦1). Note that parametric curves such as Cinclude curves having the following forms: (a) conics including lines,parabolas, circles and ellipses; Bezier, Hermite and non-uniformrational b-splines (NURBS); (b) trigonometric and exponential forms; and(c) degenerate forms like points. Additionally, note that these curveforms may be categorized orthogonally by other characteristics such asopen, closed, degenerate and composite, as one skilled in the art willunderstand.

Profile curves include curves from the following curve-type categories(2.3.1) through (2.3.5).

2.3.1. Open Curves

An “open curve” is one in which the end points of the curve are notconstrained to be coincident; e.g., the end points may be freelypositioned. Open curves are probably the most common type used by thepresent invention when defining an arbitrary collection of curves(profiles) for generating a surface (in object space), wherein thesurface is constrained to pass through the collection of curves.

2.3.2. Closed Curves

When a curve's end points match, the curve is denoted as “closed.” Thismeans that the beginning point of the curve is the same as the endingpoint of the curve. Closed curves delimit regions of, e.g., a surface,and are especially useful for setting special design areas apart. Oneexample of this is the label surface for containers (described in theDefinitions Section hereinabove); e.g., surface 66 of FIG. 2. That is, alabel surface is a region that must be of a particular surface type,denoted a developable surface, so that a label applied thereto will notcrease or tear. Each such label surface is highly constrained and isusually separated from the rest of the design by a closed curve (such acurve can also serve aesthetic purposes in the design of the container).FIG. 8 shows an elliptic region 100 blended into a cylinder 108, whereinthe closed curve 110 delimits the elliptic region. A closed curve mayoften match tangencies at end points.

2.3.3. Degenerates

Several ways exist to generate a degenerate profile. In one technique,an open curve may be of zero length, or a closed curve may enclose aregion of no area. In such cases, the result is a point that may blendwith an adjacent surface. FIG. 9 shows a point blend created fromblending between a degenerate circular disk (i.e., the point labeled S₁)and the cylinder 116 (also denoted as S₂). Accordingly a simple boss 112is created on the cylinder 116. In particular, for appropriate blendingfunctions B_(i), i=1,2, a blended surface between S₁ and S₂ can beobtained using Formula (1). Moreover, since Formula (4) can be usedinstead of Formula (1), a surface can be generated that blends between aplurality of points (i.e., degenerate profiles) and an adjacent surface.FIGS. 22 and 23 show additional blends to degenerate profiles.

FIG. 22 shows a blended surface 710 that extends between the degenerateprofile (point) 714, and the circular end 718 of the cylinder 722. Inparticular, the blended surface 710 is a blending of the isoclineribbons 726 and 730, wherein the isocline ribbon 726 is a planar diskhaving the degenerate profile 714 as its center point, and the isoclineribbon 730 has the circular end 718 as its profile. Thus, letting S₁ bethe isocline ribbon 726, and S₂ be the isocline ribbon 730 in Formula(1), the distance-like measurements (in their corresponding parameterspaces) can be equated to:

(a) the radial distance from the degenerate profile 714 on the isoclineribbon 726;

(b) the distance away from the profile 718 on the isocline ribbon 730.

FIG. 23 shows another blended surface 750 that extends between thedegenerate profile (point) 754, and the planar annulus 758 having acircular curve 760 therein (and having, optionally, a central hole 762therethrough with curve 760 as its boundary). In particular, the blendedsurface 750 is a blending of the isocline ribbon 766 (for the degenerateprofile 754), and the annulus 758 (which, e.g., can optionally be anisocline ribbon to the surface 770 wherein curve 760 is a profile).Thus, letting S₁ be the isocline ribbon 766 and S₂ be the annulus 758,the distance-like measurements (in their corresponding parameter space)can be equated to:

(a) the radial distance from the degenerate profile 754 on the isoclineribbon 766;

(b) the distance away from the curve 760 on the annulus 758.

2.3.4. Composite Curves

The novel geometric design techniques of the present invention can alsobe utilized with composite curves. Composite curves are general curveforms that include other curves as sub-curves, wherein the sub-curvesmay cross or may kink, e.g., at endpoints. In utilizing composite curvesas, e.g., profiles, the definition of a distance-like measurement for acomposite curve is important. FIG. 10 shows a composite curve 120 thatincludes two crossing sub-curves 124 and 128. However, such compositecurves can also have their sub-curves strung end-to-end.

Assuming the sub-curves C_(j), j=1,2 . . . , N of a composite curve Care parameterized and have a common parameter space, a distance formula(in parameter space) for determining a distance-like measurement D tothe pre-images of the sub-curves C_(j) is:D(p)=D _(N)(P), andD _(k)(P)=d _(k)(P)+D _(k−1)(P)−_([) d _(k) ²(P)+D _(k−1) ²(P)_(])^(1/2)  (5.5)where k=2, . . . , N and D₁(p)=d₁(P)=a distance measurement between Pand C₁, and D_(k)(p)=a distance measurement between P and C_(K). Thus,D(p) can be used as the input to a blending function, B(D), for blendingone or more surfaces to the composite curve, C.

2.3.5. Trimming Curve

The present invention allows a surface to be “trimmed,” wherein trimmingrefers to a process for constraining or delimiting a surface to one sideof a particular boundary curve (also denoted a trim curve). Inparticular, for parameterized surfaces, the pre-image of a trim curve,e.g., in the (u,v) parameter space of the surface, identifies the extentof the pre-image of the surface to remain after a trimming operation. Atrim curve may be a profile curve, and the desired trimmed surface isthat part of the original untrimmed surface that typically lies on onlyone side of the trim curve. An example is shown in FIG. 11, wherein theoriginal untrimmed surface is the generally rectangular portion 130. Therounded surface 134 is a “label” surface that is trimmed to the curve138 from the original surface 130. Note the trim profile 138 may have anassociated isocline ribbon (not shown) for one or more adjacent surfaces(e.g., surface 142) that heel to an isocline ribbon at the trimmingprofile 138. The use of isoclines for modifying the shape of suchadjacent surfaces is an important technique in creating a smoothtransition from the adjacent surfaces to a trimmed surface.

Note that the present invention may include a trimming technique tocreate a hole in a geometric object. By extruding a depression in afront surface of the geometric object through a back surface of theobject, and then trimming the front surface to exclude the correspondingportion on the back surface, a hole can be constructed that can be used,e.g., as a handle of a container.

2.4. Distance Metrics

Some techniques for computing distance-like measurements have alreadybeen provided hereinabove. In this section, additional such techniquesare described. The efficiency in computing how close a point inparameter space is to one or more particular geometric object pre-images(curves) in parameter space can substantially impact the performance ofa geometric design and modeling embodiment of the present invention. Ingeneral, for computing such distance-like measurements (these being, ingeneral, a monotonic function of the conventional Euclidean distancemetric) in parameter space between points and curves, there is atrade-off between the complexity of the curve and how efficiently suchmeasurements can be evaluated. In general, the simpler the curve, thefaster such distances can be determined. As an aside, it should be notedthat for a parameter space curve and its image curve (in object space),these curves need not be of the same computational type (e.g.,polynomial, transcendental, open, closed, etc.). Indeed, a parameterspace curve may be quite simple and still be the pre-image of acomplicated surface curve in object space. For example, the parameterspace curve corresponding to the Bezier curve 58 in FIG. 1 may be astraight line. By keeping the parameter space curve as simple aspossible, fast distance computations are possible.

2.4.1. Parametric Distance Calculations for Blending

This section describes a variety of methods for calculating adistance-like measurement (more generally, a monotonic function of theconventional Euclidean distance metric) to a number of candidateparameter space curves, wherein the methods are listed in a roughlyincreasing order of computational complexity.

Assume a blended surface is to be generated between two profile curvesP₁ and P₂, each having isocline ribbons, wherein each ribbon isparametric and has, e.g., the planar unit square [0,1]×[0,1] as thecommon parameter space for the ribbons. One distance-like functioncapable of being used for blending is a function that is dependent ononly one or the other coordinate of points represented by the coordinatepairs (u,v) in the common parameter space. That is, assuming the profilecurves P₁ and P₂ of the isocline ribbons are such that their pre-imagesare the vertical lines u=k₁ and u=k₂ for 0≦k₁≦k₂≦1, that thecorresponding distance-like functions can be D₁(u,v)=(u−k₂)/(k₁−k₂) andD₂(u,v)=(u−k₁)/(k₂−k₁). Moreover, if the pre-images are the parameterspace bounding vertical lines u=0 and u=1 (i.e., k₁=0 and k₂=1), thenthe corresponding distance-like function can be D₁(u,v)=1−u andD₂(u,v)=u, and accordingly such simple distance-like functions can becomputed very efficiently.

In order to maintain the desired simplicity in parametric distancecomputations when there are pre-images to more than two profiles forblending therebetween, three methods can be employed for computingparametric distance-like measurements. Each of the three methods is nowdescribed.

A triangular domain in parameter space bounded by, e.g., three profilecurve pre-images (that are also curves) can be parameterized withrespect to the vertices v₁, v₂ and v₃ of the triangular domain usingthree (real valued) parameters r, s and t with the additional constraintthat r+s+t=1. In other words, a point p in the triangular domain havingthe vertices v₁, v₂ and V₃ can be represented as p=r*v₁+s*v₂+t*v₃. Ther, s, t parameters are called “barycentric coordinates” and are used forthree-sided surfaces such as the surface 300 of FIG. 14 in parameterspace.

Domains in parameter space that are bounded by the pre-images of fourprofiles (denoted the four-sided case) can be a simple extension of thedomain having bounds on two opposing sides (denoted the two-sided case).In the two-sided case, if parameterized properly, only one parameter, u,need be used in the distance-like function computation. In thefour-sided case, both parameters u and v may be employed, as well astheir complements (assuming an appropriate representation such as theunit square in parameter space). Thus the distance to the four profilepre-image boundaries in parameter space can be u, v, 1−u, and 1−v (i.e.,assuming the pre-images of the profiles are u=0, v=0, v=1, u=1).

To determine barycentric coordinates for parametric space domains,assuming the pre-images of the profiles are line segments that form apolygon, the approach illustrated in FIG. 12 (illustrated for afive-sided polygon 148 having vertices v₁, v₂, V₃, V₄ and v₅) may beutilized, wherein the profile pre-images are the heavy lines labeled 149a through 149 e. To determine a distance-like function, first, stellate,i.e., make a star from, the pre-image polygon 148 by extending each ofthe sides 149 a through 149 e of the polygon until they intersect withanother extended side having a side 149 therebetween. Thus, theintersection points 150 a through 150 e are determined in the five-sidedcase of FIG. 12. Subsequently, the line segments 152 a through 152 efrom the corresponding points 150 a through 150 e to a point p in thepolygon may be constructed. The resulting distance-like measurements arethe lengths of the line segments 153 a through 153 e from p to the sides149 a through 149 e of the polygon 148. Accordingly, the distance from pto the i^(th) side 149 (i=a,b,c,d,e) of the polygon 148 is the distancealong the i^(th) line segment 153 from p to the boundary edge of thepolygon 148. Note that by dividing each resulting distance-likemeasurement by the sum of all the distance-like measurements to thepoint p, the distance-like measurements can be normalized.

2.4.2. Straight Line

A straight line is represented by the equation, au+bv=c, wherein a, b, care constants. A convenient (unsigned) distance to a line is obtained byD(u,v)=|(a,b)((u,v)−c|.  (6)For a more intuitive version that corresponds to Euclidean distance,Formula (6) can be normalized to obtainD(u,v)=|(a,b)((u,v)−c|(a ² +b ²),  (7)by dividing by the length of the gradient.

2.4.3. Conics

Conics include parabolas, hyperbolas and ellipses. The general form of aconic isAu ² +Buv+Cu ² +Du+Ev+F=0.Its unsigned distance can be computed by

$\begin{matrix}{{D\left( {u,v} \right)} = {{{\left( {u,v} \right) \cdot \begin{bmatrix}A & \frac{B}{2} \\\frac{B}{2} & C\end{bmatrix} \cdot \begin{pmatrix}u \\v\end{pmatrix}} + {\begin{pmatrix}D \\E\end{pmatrix} \cdot \left( {u,v} \right)} + F}}} & (8)\end{matrix}$

This can also be normalized by dividing through by the length of thegradient of the function to make a more suitable distance-like function,which is Euclidean in the case of the circle. Note that Farin inIntroduction to Curves and Surfaces, Academic Press, 4th ed., 1996,(incorporated herein by reference) gives the conversion between theimplicit form above and a rational parametric form. Thus, Formula (8)can be used regardless of whether the conic is represented implicitly orparametrically.

2.4.4. Polynomial Curves, Both Parametric and Implicit

Assume that a parametric curve has been converted to a Bezier form as,for example, is described in the Farin reference cited hereinabove.Vaishnav in Blending Parametric Objects by Implicit Techniques. Proc.ACM Solid Modeling Conf., May 1993 (incorporated herein by reference)gives a method to change a curve from a parametric curve to an implicitcurve numerically, wherein distance is implicitly measured in objectspace by offsetting the curve in a given direction that is based on someheuristics about how the offset is to be computed. The value of theoffset distance that forces the offset to go through the point is thedistance measurement for that point. In particular, for a Bezier curve,this distance-like measurement may be worthwhile in that it is robust(i.e., not ill-conditioned) and reasonably fast to evaluate, requiringonly two or three Newton-Raphson iterations on average, as one skilledin the art will understand. While this may be an order of magnitudeslower than computing a distance measurement of a conic representation,it is much faster than the traditional method of computing aperpendicular distance, which is also unstable.

2.4.5. Piecewise Parametric Curves

The present invention also includes a novel technique for computing adistance-like measurement on complex curves in parameter space.

Referring to FIG. 13, assume that both boundary curves 156 a and 156 bare in the unit square parametric space 158 and are piecewise parametricpolynomial curves that have corresponding sub-curves 160 a, 160 b of thesame degree n. By connecting end points of the corresponding sub-curveswith line segments 164 (i.e., degree one curves), degree n by 1 Bezierpatches 168 can be constructed in the unit square representation ofparameter space 158. Note that each patch 168 can be considered as asecond parameter space unto itself having coordinates (s,t) wherein:

-   -   (a) for two Bezier sub-curves 160 a and 160 b (denoted herein        b₁(t) and b₂(t), 0≦t≦1), each value, t₀, of t corresponds to a        line segment, ^(L)t₀, between b₁(t₀) and b₂(t₀), and    -   (b) the ^(L)t₀ line segment is parameterized by s so that        ^(L)t₀(s)εb(t₀) when s=0 and ^(L)t₀(s)εb₂(t₀) when s=1, wherein        s varies proportionally with the distance between b₁(t₀) and        b₂(t₀) when 0<s<1. Accordingly, if the distance-like measurement        between the curves b₁(t) and b₂(t)(and/or patch bounding line        segments 164) is computed in the second parameter space, then        for any (u,v) point interior to the patch, it is necessary to        find the corresponding (s,t) point relative to the boundary        curves of such a patch that can then be evaluated for        determining the distance-like measurement. Since s is the linear        parameter (corresponding to the distance of a point between the        two corresponding sub-curves 160 a and 160 b that are joined at        their endpoints by the same two segments 164), simple functions        f₁(s) and f₂(s), such as f₁(s)=s and f₂(s)=1−s, can serve as        distance functions to b₁(t) and b₂(t), respectively. Note that        the parameters u and v can both be represented as Bezier        functions of s and t. In particular, to convert from (s,t)        coordinates to (u,v) parameter space coordinates, a Newton type        algorithm may be used, as one skilled in the art will        understand.

Another approach for determining the distance-like measurement, in somecircumstances, is to evaluate such patches 168 with a “forwardalgorithm.” That is, referring to FIG. 25, an object space blendedsurface 808 that blends between, e.g., profiles 812 and 816 (havingisocline ribbons 820 and 824, respectively, to which the surface 808heels) is shown. The profile 812 has as its pre-image curve 160 a (inparameter space 158), and the profile 816 has as its pre-image curve 160b (in parameter space 158), wherein the portion of parameter space 158for surface 808 is the patch 168. An additional parameter space 828 in sand t can be considered as a pre-image parameter space for the parameterspace 158 wherein the pre-image of curve 160 a is the vertical linesegment at s=0, and the pre-image of curve 160 b is the vertical linesegment at s=1. If a sufficient dense set of points 830 denoted by “x”sin the additional parameter space 828 is used to evaluate points (u,v)in patch 168 (e.g., by determining a closest point 830), then thecorresponding points p(u,v) on a blended surface 808 can be efficientlycomputed since the distance-like functions to pre-image curves 160 a and160 b can be D₁(u(s,t),v(s,t)=s and D₂(u(s,t),v(s,t))=1−s, respectively.This approach will generate the blended surface easily and quickly.Note, if the surface 808 does not require a subsequent trimmingoperation, this method is particularly attractive.

3. Blending Programs

FIG. 17 shows a block diagram of the typical flow of design constructionoperations performed by a user of the present invention. Thus, profilehandles may be needed to construct an associated profile, and theprofile is required to construct the associated isocline ribbon, and theisocline ribbon may be required to obtain the desired shape of theassociated object (e.g., a surface), which, in turn, is required toconstruct the desired geometric model.

FIGS. 26 through 30 provide a high level description of the processingperformed by an embodiment of the present invention that enables thenovel real-time manipulation of the shape of geometric objectrepresentations so that a user can more efficiently and directly expresshis/her design intent. Moreover, it should be noted that a fundamentaltenet of the present invention is a paradigm shift away from typical CADsystems. That is, in a typical CAD system the user must supply input forchanging or modifying a geometric object and subsequently requestprocessing of the input to commence. Thus, the user receives feedbackabout his/her design at discrete user requested times. Instead, with thepresent invention, updates may be processed in real-time immediatelyupon input without the user explicitly indicating that update processingis to be performed. Accordingly, a user of the present invention canefficiently perform iterative approximations to a geometric object beingdesigned without requiring the user to precisely calculate geometriccharacteristics for substantially all portions of the object. Inparticular, this can have substantial efficiency benefits in that formany geometrically designed objects (including machined parts),substantial portions of such objects may be satisfactorily designedusing a wide range of geometrically shaped objects. Accordingly, thepresent invention allows many of these geometric objects to be designedwithout the user having to needlessly specify precision in thoseportions of the object where the precision is unnecessary.

In FIG. 26, the steps are shown for computing an interpolating curveaccording to the present invention using a one-dimensional variation ofFormula (I) discussed in Section 2 hereinabove. Accordingly, in step1004, the end points and tangent at the end points for the interpolatingcurve, C(u), to be generated are obtained. In particular, the end pointsof this curve are assigned to the variables PT1 and PT2. Additionally,direction vectors for the interpolating curve C(u) at the points PT1 andPT2 are assigned to the variables TAN1 and TAN2, respectively. Note thatPT1, PT2, TAN1 and TAN2 can be supplied in a variety of ways. Forexample, one or more of these variables can have values assigned by auser and/or one or more may be derived from other geometric objectrepresentations available to the user (e.g., another curve, surface orsolid representation). In particular, the direction tangent vectorsdenoted by TAN1 and TAN2 may be determined automatically according to aparameterization of a geometric object (e.g., a surface) upon which thepoints PT1 and PT2 reside.

In steps 1008 and 1012, the blending functions B₁ and B₂ are selected asdiscussed in Section (2.1) hereinabove. Note, however, that the blendingfunctions provided may be defaulted to a particular pair of blendingfunctions so that a user may not need to explicitly specify them.However, it is also within the scope of the present invention that suchblending functions may be specifically selected by the user. In thisregard, note that since the present invention is intended to express theuser's geometric design intent, there may be a wide variety of blendingfunctions that are acceptable since typically a user's intent is oftenadequately expressed without the user purposefully determining highlyprecise input. That is, it is believed that a wide variety of blendingfunctions may be acceptable for iterative approximation of a finalgeometric design since progressively finer detail can be provided bygenerating and/or modifying progressively smaller portions of thegeometric object being designed using substantially the same blendingfunctions. Said another way, since the present invention supports boththe entering of precise (geometric or otherwise) constraints as well asthe iterative expression of the user's intent at progressively highermagnifications, the high precision and/or small scale design featuresmay be incorporated into a user design only where necessary.

In step 1016, the interpolating curve, C(u), is computed using avariation of Formula (2) applied to a one-dimensional parameter space.An example of an interpolating curve, C(u), with points PT1, PT2, andvectors TAN1 and TAN2 identified, is illustrated in FIG. 32.

In FIG. 27, a flowchart is provided showing the steps performed whenconstructing an approximation to an isocline boundary R(u) for an object(e.g., a surface) S, wherein the points PT1 and PT2 delimit a profilecurve corresponding to the isocline ribbon boundary approximation to begenerated. In particular, the approximate isocline ribbon boundarygenerated by this flowchart is intended to approximately satisfy theisocline ribbon boundary definition in the Definitions Sectionhereinabove. More precisely, the isocline ribbon boundary approximationdetermined by the present flowchart will tend to match the isoclineribbon boundary definition for a portion of the object S between PT1 andPT2 depending on, e.g., how smooth the object is along the profile curvegenerated between PT1 and PT2. That is, the smoother (reduced curvaturefluctuations), the more likely the match. Accordingly, in step 1104 ofFIG. 27, the curve interpolation program represented in FIG. 26 isinvoked with PT1, PT2 and their respective tangents TAN1 and TAN2 forthe object (surface) S. Thus, an interpolating curve, C(u), is returnedthat is an approximation to the contour of S adjacent to this curve.

In steps 1108 and 1112, a tangent (i.e., a picket) along theparameterization of the object S at each of the points PT1 and PT2 isdetermined, and assigned to the variables PICKET1 and PICKET2,respectively. Note that typically the pickets, PICKET1 and PICKET2, willbe transverse to the vectors TAN1 and TAN2, although this need not beso. Subsequently, in steps 1116 and 1120, the isocline ribbon pointscorresponding to PT1 and PT2 are determined and assigned to thevariables, RIBBON_PT1 and RIBBON_PT2, respectively. Then, in step 1124,the curve interpolation program of FIG. 26 is again invoked with thevalues RIBBON_PT1, RIBBON_PT2, TAN1 and TAN2 to thereby generate theisocline ribbon boundary approximation, R(u). It is worthwhile to notethat in some cases, the isocline ribbon approximation bounded by theinterpolating (profile) curve C(u), the corresponding pickets (PICKET1and PICKET2), and the newly generated isocline boundary R(u) does notnecessarily form a surface. In fact, the curves, C(u) and R(u) may besubstantially coincident (e.g., if PICKET1 is identical to TAN1, andPICKET2 is identical to TAN2).

In FIGS. 28A and 28B, a flowchart for a program is provided forconstructing a more precise isocline ribbon boundary than theapproximation resulting from FIG. 27. In particular, in the flowchart ofFIGS. 28A and 28B, the program of FIG. 27 is repeatedly and adaptivelyinvoked according to the variation in the object (e.g. surface) S alongthe path of the profile curve provided thereon. Accordingly, in step1204 of FIG. 28A, a sequence of one or more markers M_(i), i=1,2, . . .N, N≧1, is assigned to the variable MARKER_SET, wherein these markersare on the surface, S, and the markers are ordered according to theirdesired occurrences along a profile curve to be generated. Note that inone typical embodiment, the markers are generally provided (e.g.constructed and/or selected) by a user of the present invention.Moreover, for the present discussion, it is assumed that the tangents tothe surface S corresponding to the markers M_(i) are tangents to Sentered by the user. However, it is within the scope of the presentinvention that such tangent vectors may be provided automatically by,for example, determining a tangent of the direction of aparameterization of the surface S.

In step 1208 of FIG. 28A, the first marker in the set, MARKER_SET, isassigned to the variable, MARKER1. Subsequently, in step 1212, adetermination is made as to whether there is an additional marker inMARKER_SET. If so, then in step 1216, the variable, INTRVL, is assigneda parametric increment value for incrementally selecting points on theprofile curve(s) to be subsequently generated hereinbelow. In oneembodiment, INTRVL may be assigned a value in the range greater than orequal to approximately 10⁻³ to 10⁻⁶.

In step 1220, the variable, MARKER2, is assigned the value of the nextmarker in MARKER_SET. Subsequently, in step 1224, the curveinterpolation program of FIG. 26 is invoked with MARKER1 and MARKER2(and their corresponding user-identified tangent vectors) for therebyobtaining an interpolating curve, C_(j)(u) between the two markers(where j=1,2, . . . , depending on the number of times this step isperformed). Then in step 1228, an isocline boundary approximation isdetermined according to FIG. 27 using the values of MARKER1, MARKER2 andthe interpolating curve, C(u), for obtaining the isocline boundaryapproximation curve, R_(j)(u).

Subsequently, in step 1240, the variable, u_VAL, is assigned the initialdefault value INTRVL for selecting points on the curves, C_(j)(u) andR_(j)(u). Following this, in step 1244, the variable INCRMT_PT isassigned the point corresponding to C_(j)(u_VAL). Subsequently, in step1245, the variable, S_PT, is assigned a point on S that is “closest” tothe point C_(j)(u_VAL) More precisely, assuming S does not fold backupon itself closer than ε>0, for some ε, a point on S is selected thatis in a neighborhood less than ε of C_(j). Note that since C_(j)(u_VAL)may not be on S, by setting the value of INTRVL so that this variable'svalue corresponds to a maximum length along the interpolating curveC_(j) of no more than one-half of any surface S undulation traversed bythis curve, then it is believed that the interpolating curve C_(j) willeffectively follow or be coincident with the surface S. Subsequently, instep 1246, a determination is made as to whether the point INCRMT_PT iswithin a predetermined distance of S_PT (e.g., the predetermineddistance may be in the range of 10⁻³ to 10⁻⁶). In particular, thepredetermined distance may be user set and/or defaulted to a systemvalue that is changeable depending upon the application to which thepresent invention is applied. Accordingly, assuming that INCRMT_PT andS_PT are within the predetermined distance, then step 1248 isencountered wherein the point R_(j)(u_VAL) on the isocline boundaryapproximation is determined and assigned to the variable, RIBBON_PT.Subsequently, in step 1252, an approximation to an isocline picket atC_(j)(u_VAL) is determined and assigned to the variable, PICKET.

In step 1254, the tangent to the surface (more generally, object) S atthe point C_(j)(u_VAL) is determined and assigned to the variable,INCRMT_TAN, this tangent being in the direction of the parameterizationof S.

In step 1256, a determination is made as to whether the vectors,INCRMT_TAN and PICKET are sufficiently close to one another (e.g.,within one screen pixel. If so, then a subsequent new point on theinterpolating curve C_(j) is determined by incrementing the value ofu_VAL in step 1264. Following this, in step 1268, a determination ismade as to whether the end of the interpolating curve, C_(j)(u), hasbeen reached or passed. Note that the assumption here is that 0≦u≦1.Accordingly, if u_VAL is less than 1, then step 1244 is againencountered, and some or all of the steps through 1256 are performed indetermining whether the isocline ribbon point approximation,R_(j)(u_VAL), is close enough to the actual ribbon point astheoretically defined in the Definitions Section hereinabove.

Referring again to step 1246, note that if INCRMT_PT is not close enoughto S, then an interpolating curve more finely identified with the actualpoints of S is determined. That is, the point, S_PT, is made into amarker and inserted into MARKER_SET, thereby causing new interpolatingribbon curves, C_(j)(u) and R_(j)(u) to be generated that will deviateless from S (assuming S is continuously differentiable). That is, step1272 is performed wherein a marker is generated for the point, S_PT, andthis new marker is inserted into MARKER_SET between the current markervalues for MARKER1 and MARKER2. Subsequently, the marker currentlydenoted by MARKER2 is flagged as unused (step 1276), and in step 1280,the most recently constructed interpolating curve C_(j)(u) and anyassociated ribbon boundary curve R_(j)(u) are deleted. Then, step 1220and subsequent steps are again performed for determining newinterpolating and ribbon boundary curves, C_(j)(u) and R_(j)(u).

Note that steps 1272 through 1280 and step 1220 are also performed if instep 1256, INCRMT_TAN and PICKET are not determined to be sufficientlyclose to one another in the object space of S.

Referring again to step 1268, if the end of the interpolating curve,C_(j)(u), has been reached or passed, then it is assumed that C_(j)(u)is sufficiently close approximation to points on S (between MARKER1 andMARKER2), and R_(j)(u) is insufficiently close to an isocline ribbon forthese points on S. Thus, if there are additional markers wherein aninterpolating curve C_(j)(u) and corresponding ribbon approximationR_(j)(u) has not been determined, then the next pair of consecutivemarkers (of the marker ordering) in MARKER_SET is determined and variousof the steps 1220 and beyond are performed. That is, in step 1284,MARKER1 is assigned the value of MARKER2, and in 1288, a determinationis made as to whether there is a next unused marker in MARKER_SET. Ifso, then variations of the steps 1220 and beyond are performed asdescribed above. Alternatively, if all markers have been designated asused, then in step 1292 the resulting curves C_(j)(u), R_(j)(u), foreach used j=1,2, . . . , are graphically displayed and stored forsubsequent retrieval. Note that the profile curves C_(j)(u) may beoptionally reparameterized so that these curves may be parameterizedcollectively as a single curve, {tilde over (C)}(u), with {tilde over(C)}(0)=C₁(0) and {tilde over (C)}(1)=C_(N)(1).

FIGS. 29 and 30 provide high-level descriptions of flowcharts formodifying one or more surfaces (more generally geometric objects) bymodifying isocline handles, ribbon tangents, and their associatedisocline ribbons. In particular, for simplicity, the flowcharts of thesefigures assume that there is a composite surface, S₀, that is provided(at least in part) by one or more subsurfaces, S_(i), i=1,2, . . . N,N≧1, where these subsurfaces S_(i) are connected to one another (e.g.,patched together) along common boundaries so that S₀ does not havedisconnected portions. Accordingly, given such a composite surface, S₀,the flowcharts of FIGS. 29 and 30 can be described at a high level asfollows. In FIG. 29, an isocline handle and/or a ribbon tangent havingat least one geometric characteristic (e.g. length, direction,curvature, etc.) to be changed is determined along with the subsurfacesS_(i) that are to be modified to reflect the isocline handle and/orribbon tangent changes. Subsequently, in the flowchart of FIG. 30, themodifications to the subsurfaces are computed and displayed in real-timeas the user enters the modifications to the selected isocline handleand/or ribbon tangent. Note that the computing of surface (moregenerally geometric object) modifications in real-time has notheretofore been feasible for surfaces in higher dimensional geometricobjects in that the computational overhead has been too great.Accordingly, the present invention has reduced this overhead byproviding a novel technique of computing blended surfaces which is veryefficient and which generates surfaces that are fair.

The following is a more detailed description of FIGS. 29A and 29B. Instep 1400, if there are not profiles and isocline ribbons correspondingto the entire boundary of each subsurface S_(i), then make profiles andisocline ribbons that approximate the entire boundary of each subsurfaceS_(i). Note that this may be performed using the program of theflowchart of FIG. 28. In step 1404, the isocline handles and ribbontangents corresponding to markers on the surface SO are graphicallydisplayed to the user. In step 1408, a determination is made as towhether the user has requested to add one or more additional isoclineribbons to the surface S₀, or extend an existing isocline ribbon havingits profile curve on S₀. If the user has made such a request, then step1412 is performed to assure that in addition to any other markers addedby the user, markers are added: (a) whenever a profile contacts aboundary of a subsurface S_(i), and (b) so that profile curves will beextended in a manner that terminates each one on a boundary of asubsurface S_(i). Moreover, additional markers may be also added atintersections of curve profiles. Thus, for these latter markers, theremay be two distinct ribbon tangents associated therewith (i.e., one foreach subsurface).

Subsequently, in step 1416, the program of FIG. 28 is invoked with eachS_(i), i=1,2, . . . N for thereby obtaining the desired additionalprofiles and isocline boundaries. As an aside, note that FIG. 28 needonly be invoked with the subsurfaces S_(i) to which new markers areadded.

In step 1420 following step 1416, all newly added isocline handles andribbon tangents are displayed. Note that in some embodiments, only theisocline handles are displayed initially, and the user is able toselectively display the ribbon tangents as desired.

Subsequently, in step 1424, a determination is made as to whether theuser has requested to add one or more additional markers within existingprofiles. If so, then the additional new markers are added and at leastthe corresponding new isocline handles are determined for these newmarkers. As an aside, note that in one embodiment of the presentinvention, when a new marker is added to an existing profile, theprofile will change somewhat since it is now exactly identical to thesurface S₀ at another point and the interpolating curve generated (viaFIG. 26) between consecutive markers of a profile is now generated usingthe newly added marker. Accordingly, a profile with one or moreadditional markers should, in general, conform more closely to the shapeof the adjacent portions of the surface S₀.

Subsequently, in step 1432, the additional new markers and optionally,their corresponding isocline handles and ribbon tangents, aregraphically displayed to the user.

Note that it is not necessary for steps 1408 through 1420, and steps1424 through 1432 to be performed sequentially. One skilled in the artof computer user interface design will understand that with event drivenuser interfaces, the processing of each new marker can be performedindividually and displayed prior to obtaining a next new marker locationfrom the user. Thus, consecutive executions of the steps 1408 through1420 may be interleaved with one or more executions of the steps 1424through 1432.

In step 1436, a determination is made as to whether an isocline handleand/or a ribbon tangent is selected by the user for modification. Notethat the identifier, ISO, will be used to denote the isocline handleand/or the ribbon tangent to be modified.

In step 1440, the marker corresponding to ISO is determined and accessthereto is provided via the variable, MRKR. Subsequently, in step 1444,the collection of one or more subsurfaces S₁, . . . S_(N) adjacent toMRKR are determined and access to these adjacent subsurfaces is providedby the variable, ADJ_SURFACES.

In steps 1448 through 1460, boundary representations of portions of thesubsurfaces, S_(i), adjacent MRKR are determined (step 1452) andinserted into a collection of surface boundary representations denotedMOD_SET (step 1456). In particular, for each of the subsurfaces inADJ_SURFACES, a data representation of the boundary of the smallestportion of the subsurface that is adjacent to MRKR that is bounded byisocline ribbons is entered into the set, MOD_SET.

Finally, in step 1464, the program of FIG. 30 is invoked for modifying,in real-time as the user modifies ISO, the portion of S₀ within theboundary representations contained in MOD_SET. In particular, theprogram of FIG. 30 is invoked with the values, MRKR and MOD_SET.

In the flowchart of FIG. 30, the high-level steps are shown formodifying in real-time the surface portions identified by the surfaceboundary representations in MOD_SET, wherein these surface portions areadjacent to the marker, MRKR. Accordingly, in step 1504, the first(next) modified version of the isocline handle and/or ribbon tangentcorresponding to the marker, MRKR, is obtained and assigned to, ISO.Subsequently, in step 1508, all isocline ribbons containing the modifiedisocline handle and/or ribbon tangent of ISO are regenerated to reflectthe most recent modification requested by the user. Note that this isperformed using the one-dimensional version of Formula (I), andmodifying each such isocline ribbon along its extent between MRKR andthe adjacent markers on each isocline ribbon containing MRKR.

Subsequently, in step 1512, the first (next) boundary representation inMOD_SET is assigned to the variable, B. Then in step 1516, the set ofisocline ribbons for the (profile) boundary segments contained in B areassigned to the variable, R. Note that R includes at least one isoclineribbon containing the marker, MRKR.

In step 1520, a blended surface is generated that is delimited by theprofiles of the isocline ribbons of R. The formula used in this step issimilar to Formula (4). However, there are additional functions,Q_(i)(p), provided in the present formula. Note that, in general, theportion of a parameter space used in generating the surface, S(p), ofthis step may have two, three, four, five or more sides (profilepre-images) that also have isocline ribbon pre-images. Thus, for thetranslation functions, Q_(i)(p), for each isocline ribbon R_(i) of R(wherein for the points p in the parameter space that are in theinterior, 1, to the pre-images of the profiles, P_(i), for the isoclinesR_(i) of R) it is desirable that these points p be translated intopoints in the parameter space for R_(i) so that a corresponding point inthe object space of the isocline ribbon R_(i) can be determined and usedin the blending function of the present step. Note that the translationfunctions, Q_(i)(p), preferably satisfy at least the followingconstraints:

(a) Q_(i)(p) is a continuous function for continuous surfaces;

${\underset{p\rightarrow{Q_{i}^{- 1}{({u,0})}}}{LIMIT}\left( {Q_{i}(p)} \right)} = \left( {u,0} \right)$That is, when a sequence of points in 1 converges to the pre-image ofthe profile point P_(i)(u), then Q_(i)(p) converges to the isoclineribbon parameter space point (u,0).

Subsequently, in step 1524, the surface S(p) is displayed, and in step1528 a determination is made as to whether there is an additionalboundary representation in MOD_SET for generating an additional blendedsurface S(p). If so, then step 1512 is again performed. Alternatively,if there are no further boundary representations, then in step 1532, adetermination is made as to whether there is an additional usermodification of the isocline handle and/or ribbon tangent correspondingwith MRKR. If there is, then at least the steps 1504 through 1528 areagain performed. Note that the steps of FIG. 30 can be sufficientlyefficiently performed so that incremental real-time changes in theisocline handle and/or ribbon tangent for MRKR designated by the usercan be displayed as the user continuously modifies this isocline handleand/or ribbon tangent.

4. A Geometric Design User Interface

The general principles described above form a basis for a novel userinterface for computer aided geometric design.

In one user interface embodiment for the present invention, a userinterface may be provided for defining isoclines.

Using such an interface, a designer may, for example, require that anisocline be perpendicular to a given light direction along an entireprofile curve so as to create a reflection line, as one skilled in theart will understand. More generally, the novel user interface may allowfor various constraints to be input for generating isocline ribbons,isocline handles and/or ribbon tangents that satisfy such constraints.In particular, the user interface allows for global constraints such aslight direction, curvature, tangency, level contours, dihedral anglefunctions with a plane, etc., as one skilled in the art will appreciate.

In one embodiment of the user interface, the user will start with agiven geometric object, for example a cylinder. The user may theninscribe a profile curve on the cylinder by placing markers at variouspoints on the cylinder. The profile tangents and/or isocline handles maybe defaulted by adopting the slope information from the cylinder. Forexample, at each marker, the profile tangents are in the plane tangentto the cylinder at the marker.

The user may then select and modify the markers, add additional markers,and/or modify the position and the direction of the isocline handlesand/or ribbon tangents. As the isocline ribbon is accordingly modified,the cylinder (more generally, geometric object) will reflect the changesin the modification of the isocline ribbon. Additional profiles andmarkers may be added in this manner until the desired shape of thegeometric object (derived from the cylinder) is obtained. An example ofthese steps is illustrated in the flowchart of FIG. 31. That is, theuser selects a graphically displayed surface (more generally, geometricobject) in step 1904. Subsequently, in step 1908, the user constructs aprofile curve on the selected surface (object).

Subsequently, in step 1912, an isocline ribbon (or at least the isoclineboundary) is generated for the profile. Note that this ribbon/boundarycan, if desired, be generated substantially without additional userinput. That is, an isocline ribbon/boundary may be generated from thetangency characteristics of the surface upon which the profile resides.In particular, for a parametric surface (more generally geometricobject), the parametric tangents on the surface at points on the profilecan be utilized to generate an isocline ribbon/boundary for the profile.Thus, surface neighborhoods on one side of the profile curve may be usedto determine a first isocline ribbon/boundary for a first surface havingthe profile, and if the profile is on the seam between the first surfaceand a second surface, then surface neighborhoods on the other side ofthe profile may be used to determine a second isocline ribbon/boundary.

Additionally, note that other surface characteristics may be preservedin an isocline ribbon/boundary. For example, in addition to preservingthe parametric tangents at profile curve points, the isoclineribbon/boundary may also optionally preserve the surface characteristicssuch as curvature, high order (>=2) derivative continuity with thesurface. Note, however, it is within the scope of the present inventionthat further surface characteristics can be preserved in the isoclineribbon/boundary.

In step 1916, the generated isocline ribbon/boundary may be used tomodify the surface(s) having the profile curve as discussed hereinabovewith reference to the programs of the flowcharts of FIGS. 29 and 30.

In some embodiments of the user interface, an operation is provided tothe designer wherein a common boundary between two object space surfacescan be selected and the operation automatically forces the surfaces tojoin at a higher order continuity criteria (e.g., curvature continuity)than that of tangent plane continuity. For example, a higher ordercontinuity constraint imposed on an isocline ribbon deriving from one ofthe surfaces at the common boundary, can be used to similarly constrainan isocline ribbon for the other surface having the common boundary.Accordingly, this operation helps alleviate the so-called “Mach band”effect in which the human eye detects discontinuities in curvature insome instances.

Other user interface operations provided by the present invention are:

-   -   (a) Rounderizing, which is a tweaking operation that modifies an        existing surface to round off pointed edges, or to create darts        (i.e., surfaces that are smooth except at a single point, where        the surface kinks) that dissipate sharp edges. Such operations        can be performed using the present invention by positioning        profile curves on the surface on opposite sides of a sharp edge        and blending smoothly between the profiles (e.g., using        Formula (1) as described in Section 2.3.5; and subsequently        eliminating the surface in between the profile including the        sharp edge.    -   (b) Embedding, which is an iterative user interface procedure        that can take one finished model, scale it, and perhaps rotate        or otherwise deform it to fit into part of another model.        4.1. Defining the Isocline via Markers, Profiles and the User        Interface

Explicit profiles are the profile curves that express a designer'sintent. Explicit profiles may be unconstrained (freeform) or partiallyconstrained (trim). Implicit profiles may be visible boundaries betweensurface patches caused, for example, by a surface discontinuity (i.e., akink or curve defined between an end surface of a cylinder and thecylindrical side thereof).

Implicit profiles are created automatically when the user introduces,e.g., a surface discontinuity. All profiles in a model are eitherexplicit or implicit.

4.1.1 Creating Markers

Profile markers and handles are created in the following ways:

-   -   A. Markers are automatically created at the ends of new implicit        and explicit profiles.    -   B. Inserted by a designer (e.g., by double clicking at a point)        on an explicit profile. To the designer, he/she is inserting a        point on the profile. The newly placed marker only minimally or        not at all changes the shape of the profile in the profile        segment containing the new marker. Subsequently, profile and        isocline handles are determined according to the shape of the        profile and surface(s) attached at the new marker.

A marker may be identified with a plurality of coincident points on thesame profile (e.g. a profile that loops back and attaches to itself).Such an identification of the marker with the plurality of profilepoints cannot be broken, except by deletion of the marker. In the caseof two or more profiles meeting at a common point having a profilemarker, such profiles each have a marker at the common point and themarkers are constrained to maintain coincidence so that moving onemarker will move both.

Profile markers inserted by the designer may be inserted for providingprofile handle points, or for setting specific isocline values. Notethat a profile handle point may have a set of constraints on itsisocline handles; i.e., isocline handle may inherit value(s) by aninterpolation of the nearest two adjacent isocline handles.

4.1.2. Viewing Markers and Profiles

Profile and isocline handles may have various constraints placed uponthem, wherein these handles may be displayed differently according tothe constraints placed upon them. In particular, the followingconstraints may be placed upon these handles:

(a) constrain a handle to a particular range of directions;

(b) constrain a handle to a particular range of magnitudes;

(c) constrain a handle to lie in a plane with other handles;

(d) constrain a handle to a particular range of curvatures;

(e) constrain a handle with a transform of another handle, e.g.,identical rotations and/or translations.

The designer can choose to display the constraints through a displayrequest for the properties of geometric objects. In one embodiment,different colors may be displayed for the different types of constrainedprofile markers. For example, handles having no variability (alsodenoted herein as “fully constrained”) may be displayed in blue.In some embodiments of the user interface, vectors are “grayed out” thatare constrained to thereby demonstrate to the designer that such vectorscannot be changed. For example, in one embodiment, fully constrainedhandles are typically grayed out.

4.1.3 Connecting Profiles Together

In one embodiment of the user interface, it supports the linkingtogether of two or more profiles that intersect at the same X,Y,Zlocation. Such intersection points are denoted “tie points,” when theparameterization of the point on each profile is invariant under profilemodifications. Note that such tie points may or may not have markersassociated therewith. When such a tie point is modified, allcorresponding profile curve points associated therewith at the tie pointare modified as a group. Such a tie point may be an endpoint of aprofile or an internal (i.e. “knot”) point.

Alternatively, a profile marker of a first profile may be constrained tolie within the object space range of a second profile (either implicitlyor explicitly). For example, referring to FIGS. 42A and 42B, the userinterface may provide the user with the capability to slide a profilemarker 2002 (contained on a first profile 2003 and a second profile2004) along the second profile 2004 for thereby changing the profile2004 of FIG. 42A into the profile 2004 of FIG. 42B when the marker 2002is slid along the profile in the direction of direction arrow 2006. Sucha slidable marker 2002 is denoted as a “slide point.”

Profile intersections are either “slide” or “tie” points. Moreover,these different types of points may be distinguished graphically fromone another by different colors and/or shapes. Note, if a profile slidesalong another profile, and the isocline ribbon for the sliding profileis used to compute a blended surface, S, then S will be recomputed.

4.1.4 Creating Markers and Profiles

The user interface may support the creation of a profile curve in anumber of ways:

A. Sketch the profile on the surface similarly to the data driventechnique of FIG. 28, wherein additional markers may be provided fortying the profile to the surface within a predetermined tolerance.Alternatively, in a second embodiment, a profile can be sketched acrossone or more surfaces by having the user select all markers for theprofile. Note that in either case, a profile may be sketched across oneor more surfaces. Moreover, in the second embodiment, the user interfacesupports the following steps to successfully create a profile.

(A1) Marker points are specified which lie on the surface(s). A fittedcurve is generated through the points in parameter space and then thisfitted curve is evaluated to obtain a corresponding image curve inobject space.

(A2) For each surface that the new profile crosses, the user may specifythe profile type (freeform or trim) for the portion of the profilecrossing the surface. A profile sketched on a surface either trims oneside of that surface or splits that surface into two new surfaces.Accordingly, note that any (non-degenerate) profile that enters theinterior of a surface must cross the surface's boundary at an enteringpoint and an exiting point. That is, the two surfaces along a commonprofile boundary are always linked to the profile, wherein, with respectto this profile, these surfaces may be characterized as follows: (a) onesurface is a trim surface and one is a non-trimmed surface (also denoteda “freeform surface”), or (b) two freeform surfaces.

B. Copy a Profile: A designer selects a profile to copy. The profile iscopied to a buffer (denoted a clipboard). The user then selects the modeof copy (examples: keep profile handles, or adapt profile handles to thegeometry where the profile is to be copied). The user selects where toposition the new profile (which may be additionally scaled, rotated,and/or mirrored, etc). The user selects a location for the new profile.Note that the new profile can be constrained by constraints on theoriginal profile. For example, the new profile may be mirrored about aplane from an existing profile, such that any change to the originalprofile will result in a change to the copy.

When a new profile is created, profile markers are automaticallygenerated at both ends of the new profile. Profile handles and isocline(ribbon tangent) handles are derived from the geometric characteristicsof the surfaces that the new profile splits.

4.1.5 Modifying Markers and Profiles

Modification of markers and/or (profile, isocline) handles is dependenton the constraints placed upon the markers and/or handles. They may beconstrained in one of the following ways:

A. Interactive modification by selecting a handle (profile or isocline)at a particular marker, m, and moving the handle endpoint so that it isconstrained to lie in a normal plane, i.e., either a plane normal to asurface having a profile containing m, or a plane defined by theisocline and profile handles of the profile at m. A pop-up propertysheet is also available for the user to key-in specific numerical valuesfor magnitude and angle for modifying a selected profile and/or isoclinehandle.

B. For markers that are constrained so that their pre-images lie withinthe pre-image of a profile, the marker will may slide along such aparent profile via, e.g., interactive modification by dragging suchmarker points. Note that for positional unconstrained markers, themarker points may be moved freely (i.e., under a user's direction andwithout other restrictions) along the parent profile(s) upon which suchmarkers reside. Additionally, note that the user can select multipleprofile marker points by clicking on each, or by selecting all markerswithin a designated region (e.g., bounding rectangle). Accordingly, theuser is able to move a display pointing device (e.g., a mouse) whichwill then cause all of the selected markers to uniformly move in acorresponding direction to the movement of the display pointing device.However, movement of the markers depends on the constraints set on thesemarkers. For example, a constrained marker will only move within thelimits placed upon it. Thus, if a first selected marker moves onlywithin a first profile and a second selected marker only moves within adifferent second profile oriented orthogonally to the first profile,then depending on the direction of movement desired, one of thefollowing occurs:

-   -   (i) the first marker is able to move, but the second marker can        not;    -   (ii) the second marker is able to move, but the first marker can        not;    -   (iii) both the first and second markers are able to move;    -   (iv) neither marker may be able to move.

C. Marker and handle constraints may be set by default rather thanexplicitly by the user. A pop-up property display form allows the userto set or remove specific constraints.

D. Additional constraints on profile and/or isocline handles may be setthat are dependent on the characteristics of other geometry. Forexample, profile and isocline handles can be constrained to be normal orparallel to a selected reference plane. Note that the position of aprofile marker also can be constrained to be dependent oncharacteristics of other geometry. For example, a marker can beconstrained to lie on a parting plane. That is, a plane of front/backsymmetry for designing a bottle. Another example of these constraints isin generating symmetric designs, i.e., a profile marker copy that isreflected about a parting plane will be constrained to be symmetric tothe parent profile marker.

E. Surfaces adjacent to a profile may have to satisfy either C0, C1, orC2 continuity, wherein C0 is positional, C1 is tangency, and C2 forcessmooth surface blends. One constraint that can be set on a marker is toforce C1 continuity between surfaces surrounding the marker bymaintaining equal length tangent vectors interior markers.

Freezing (eliminating the ability to modify) the profile and isoclinehandles at a marker will cause the profile segment containing the markerto rebuild based on the profile handles of the next two closest markers,effectively changing this marker's handles to reflect the curve built bythe two markers on either side.

For the most part, modifying profiles is a function of user interfacetechniques for modifying the profile marker positions and handles thatcontrol the shape of the profile. The following are examples of suchuser interface techniques.

A. Direct method: Profiles are modified directly on an object space (3D)model. This is done by modifying the profile markers and handles thatmake up the profile.

If the designer modifies a trim profile, the profile always lies withinthe parameter space of the surface it is trimming. That is, the trimprofile needs to be modified in the context of its original, overbuiltsurface within which it is embedded. Selecting a trim profile (or one ofits components) to modify causes the overbuilt, construction geometry tobe highlighted. It remains highlighted while the user is modifying thetrim profile.

A designer may have the option to turn on profiles and modify them usingthe direct method. For example, modifying a profile that defines anoverbuilt surface will cause the overbuilt surface to be updated. Sincethe profile that trims this overbuilt surface is constrained to liewithin the parameter space of the surface, the trim profile is alsorecomputed.

B. Design Ribbon method: This method is used to modify a specifiedregion of a profile. It allows, for example, the user to simplify thedesigner's interactions by modifying a profile in one view that iscomplex in another view. The designer identifies two markers that lie onthe same profile. The profile segment(s) between the two markers areextruded in at least one graphical view of the profile, thereby creatinga design ribbon (not to be confused with an isocline ribbon). The designribbon is a simple extruded surface (i.e., a curve which is swept alonggiven directions to generate a surface; for example, for markers at theends of a profile, offset the corresponding profile handles by theircorresponding isocline handles to obtain boundary handles andinterpolate an isocline boundary, e.g., by a lower dimensional versionof Formula (2); the surface having a perimeter consisting of theprofile, the isocline handles, and the isocline boundary defines the newextruded surface). The three-dimensional profile segments identifiedbetween the markers always lie within the pre-image of this designribbon. The user modifies the profile in one of two ways:

-   -   (B1) Modify the two-dimensional driving curve from which the        design ribbon was extruded, and which is instanced at the end of        the ribbon, wherein, by default, this curve is a two-dimensional        representation of the three-dimensional profile segments        defining the design ribbon. The user may “simplify” the driving        curve by selecting a subset of the two-dimensional points. Each        time the user modifies a driving curve point, the ribbon is        updated and the three-dimensional profile is modified to lie        within the parameter space of the modified ribbon. Operations on        the driving curve include any of those listed in the Profile        Marker section (point/slope modification, insert, delete, etc.).    -   (B2) Modify the two-dimensional profile points within the design        ribbon. The user directly modifies the two-dimensional profile        in a view that is perpendicular to the primary view. The        two-dimensional points always lie in the parameter space of the        ribbon. Operations include any of those listed in the Profile        Marker section (point/slope modification, insert, delete, etc.).

Only one design ribbon may exist per surface per profile segment. Designribbons may be created, modified, and deleted. Once they are created,they are persistent, i.e. they remain unmodified until a designermodifies the same segment at a later point in time. A design ribbon isdisplayed only when a designer is modifying it. A single profile mayhave multiple ribbons corresponding to multiple surfaces containing theprofile.

Note that modifying a profile using the direct method deletes any designribbons spanning the points being modified. This invalidates the ribbonand requires a designer to re-specify the ribbon.

-   -   C. Move Profile(s): The designer selects and moves two or more        profiles in unison. That is, this user interface command selects        all of the profile markers on a profile (or segment thereof) and        moves them together as a unit.    -   D. Merge Profiles: The designer may sketch a new profile and        attach it to an existing profile so that an endpoint of each        profile is coincident. Additionally, the designer may specify        which segment or segments of the existing profile to delete.        Subsequently, the new profile and the remaining connected        portion of the existing profile having the coincident end point        with the new profile are merged. Note that merging profiles        causes the set of the respective profile handles, isocline        handles and ribbon tangents for each of the two coincident        endpoints to be combined into a single such set.    -   E. Split profile: Split one profile into two at a single        point, p. An endpoint of each of the two new profiles is        constrained to be coincident at p.

4.1.6 Deleting Markers and Profiles

Deleting a profile marker is always possible, except at the endpoints ofa profile. However, in some embodiments of the present invention mayneed to replace a marker with a marker having constraints if it isneeded for maintaining smooth patches. Note that a new constrainedreplacement marker may or may not be in the same location as theprevious marker.

If an entire profile is deleted, then the user interface both highlightsany dependent geometric object(s) and requests user confirmation beforedeleting the profile and the dependent geometric object(s). Accordingly,note that the present invention retains sufficient dependencyinformation regarding dependencies between geometric objects in a modelso that for modifications of an object that is used for deriving otherobjects, appropriate additional modifications can be performed on theseother objects automatically.

4.1.7 Profile Markers and Handles

Note that there are typically two profile handles, two isocline handlesand two ribbon tangents for a profile marker, i.e., a profile handle, anisocline handle and ribbon tangent per surface on each surface havingthe profile as a boundary curve. However, there may be more handlesassociated with a profile where several profiles converge or fewer ifthe profile is the edge of a surface.

4.2 Isoclines and the User Interface

The slope of an isocline handle controls surface tangency at a markerand at a surrounding portion of the profile containing the marker. Themagnitude of an isocline handle controls the fullness of the dependentsurface. That is, how much the surface bellies out. An isocline handlemay be constrained to be offset from another isocline handle (i.e. −10degrees from other side). An isocline handle can be calculated at anypoint along a profile (by inserting a marker on the profile).

4.2.1 Creating an Isocline Handle

The user interface supports the constraining of isocline handlesrelative to one another. Such handles can be forced to always betangent, of equal magnitude, or offset by some amount. In one embodimentof the present invention, the user interface provides a pop-up menu todisplay and change isocline handle constraint values, such as length anddirection.

4.2.2 Modifying an Isocline Handle

If the user slides a profile marker along a profile, the user may fixthe isocline handle for the marker, thereby causing the surfacesadjacent to (and dependent upon) the profile to change or have theisocline handles interpolated between the nearest two isocline handleson the profile (which case implies that the dependent surfaces are notaffected)

4.2.3 Deleting an Isocline Handle

The user interface supports the deletion of isocline handles. Thefollowing steps apply when deleting an isocline.

4.3 Special Geometric Objects and the User Interface

The present invention provides for the creation and manipulation of anumber of specialized geometric object types that can substantiallyfacilitate the design of objects such as containers.

4.3.1 Label Surfaces

A label surface is a special case of a trimmed surface. The special caseaspects of a label surface are:

-   -   (i) there is a “watershed” profile that runs from the bottom to        the top of the label;    -   (ii) there are label curves between which the corresponding        label surface is ruled (e.g., label curves 132 of FIG. 11);    -   (iii) there is a boundary (trim) profile (e.g., trim profile 138        of FIG. 11).

The key difference that makes a label surface different from othertrimmed surfaces is that the original surface (from which the labelsurface is trimmed) is a ruled surface. In particular, the label surfacedefining curves are constrained such that a ruled surface is maintainedwithin the boundary of these defining curves.

Note that other surfaces may be blended to a trim profile for a label,but the trim profile can only be modified in a manner that insures thatit bounds a ruled surface.

In one embodiment of the present invention, a two dimensional “rolledout” representation of the label surface can be generated. That is, thesurface can be associated in a one-to-one fashion with the plane byrolling it flat. Such a representation simulates a label surface inwhich a designer can thereon create a piece of artwork that cansubsequently be wrapped on a container.

4.3.1.1 Creation of a Label Surface

To create a label surface, an overbuilt surface to be trimmed must be aruled, approximately developable surface, i.e., a ruled surface in whichall surface normals on any ruling are parallel. Subsequently, the userthen follows the normal trim surface steps; i.e., sketching a profile onthe ruled surface, generating the (sub)surfaces on both sides of theprofile (i.e., the label surface and the portion of the surface to betrimmed away), trimming the label surface and blending other surfaces tothe trim profile.

Note that the trimmed away surface portion is hidden from normal viewing(i.e., it is no longer a part of the visible model).

The following procedure may be provided for generating a label surface.At a high level, the steps for this procedure are:

-   -   (i) Make sure the surface is ruled. That is, the user interface        supports automatic modification of user selected profiles so        that these profiles satisfy 4.3.1(i) and (ii). In particular, to        perform this step, the following substeps are performed:    -   (ii) The user sketches a boundary profile on the surface        defining the bounds of the label;    -   (iii) Construct a graphical representation of a label (i.e., a        ruled surface having text, artwork, and/or designs thereon);    -   (iv) Allow the user to graphically apply the label        representation to the label surface (or a representation        thereof). In particular, the user interface for applying the        label representation may automatically attach the label        representation to the label via a grouping type of operation so        that the label representation maintains its position on the        label surface during, e.g., label surface rotations,        translations, scaling operations, etc.;    -   (v) Allow the user to undo the design when the label surface        and/or the label is not satisfactory.        4.3.1.2. Modification of a Label Surface

Modification of the label surface components is somewhat different thanthose of a trimmed surface.

The portion of the watershed profile that is a straight line segment isconstrained to remain straight.

The boundary opposite of the watershed (the “other side, of the parentruled surface) cannot be modified. It is simply a straight line segmentbetween the top and bottom boundaries. The top and bottom boundaryprofiles can be modified. They are constrained so no additional freeprofile markers can be inserted on them. Also, the profile marker at theend away from the watershed is constrained to move only to maintain aruled surface. It can be extended (extrapolated along the samecurvature) and the angles at its endpoints can be adjusted—again, aslong as it maintains a ruled surface.

4.3.1.3. Deleting of a Label Surface

Deleting a label surface removes the constraints on all of the profilesused in creating the label surface. Additionally, all of theconstruction geometry for the label surface that is invisible to theuser will also be deleted. The constraints for maintaining a ruledsurface will also be removed. Thus, the remaining geometric objects arethen freed from the label surface constraints, and can be modified inways not previously available.

4.4. Hole Tool User Interface

The present invention may also provide a user with a novel computationalmethod that helps the user add a hole to a geometric model (such as foradding a handle to grasp to a non-handled bottle). The informationrequired to add a hole to a model using this procedure includes: a loopof profile segments on a front surface, a loop of profile segments on anopposite back surface, the type of each profile in a loop of profiles(freeform or trim), and optional profile(s) to shape the interior of thehole.

4.4.1. Creation of a Hole

The hole creation tool guides the user through a series of steps to adda hole. FIGS. 22A–22C illustrate the procedure for creating a hole 600(FIG. 22C) on a geometric object 604 using the present invention. Thecorresponding steps performed for creating the hole 600 are as follows:

-   -   (a) Sketch a profile loop 608 on the front surface 612;    -   (b) Sketch a profile loop 616 on the back surface (optionally        project the profile 608 to the back surface).

Note that isocline handles are automatically placed on both profiles 608and 616.

-   -   (c) If one or more of the profiles for one of the profile loops        608 and 616 are freeform profiles, then the user may add new        profiles (to complete such a profile loop), and/or profiles        merge corresponding to such a loop whereby these profiles are        constrained so that they are utilized as if they were a single        profile. Accordingly, once the profile loops are constructed,        then surfaces may be skinned between the loops to thereby        replace the original surface occupying the hole.

If a trim profile loop is specified, the surface region inside theprofile loop is trimmed.

If specified, the hole creation procedure uses additional profiles toplace and shape surfaces on interior boundaries for the hole. Otherwise,surfaces are skinned automatically between the front and back profileloops.

4.4.2. Modification of a Hole

Modifying a hole is a function of modifying profile markers and handlesthat make up the geometry of the hole.

4.4.3. Deleting of a Hole

Deleting a hole is also a function of deleting the components that makeup the geometry of the hole; i.e., profiles and other geometry for thehole.

4.5. Smoothness Considerations via the User Interface

We consider the order of transition between adjacent surfaces (whichmeet at the profiles). This section does, however, contain some broaderimplications for the general theory in Section 2.

4.5.1. Continuous Profiles via the User Interface

Given two profile curves that intersect, derivative continuity across anintersection point may be assured if several conditions are met:

-   -   (a) an end point of one profile is coincident with an end point        of the other profile (positional continuity);    -   (b) the blending functions B_(i) used in generating the profiles        (as per FIGS. 26 and 27) are equal at the intersection marker;        and    -   (c) the profile handles at the intersection marker are collinear        and equal length.

Tangent directional continuity is a weaker condition that can besatisfied if condition (c) above is changed to:

(c*) the profile handles at the intersection marker are only colinear.The magnitudes of the profile handles may differ in this case.

The designer may intentionally produce a kink at a marker by breakingthe collinearity of the two profile handles at the intersection marker.This means that the two profile handles do not have a common direction.

4.5.2. Continuous Surfaces

The notion of tangent plane continuity between surfaces may be definedas follows: for each point p of a boundary between two surfaces S₁ andS₂, the tangent plane, T₁(p), of S₁ at p is identical to the tangentplane, T₂(p), of S₂ at p. To achieve tangent continuity between surfacesacross a profile boundary therebetween, it is necessary that theisocline handles (for each of the surfaces) at each marker on theprofile boundary lie in a common plane with each other and the profilehandle at that marker. If this is not done then a kink in the surfacealong the profile will be created.

Note that when there are two or more surface patches to be generatedwherein these patches must be constrained to meet at a common markerpoint, p, the present invention may automatically generate isoclinehandles, denoted “common direction handles.” That is, for each profile,P (having p) used in defining one or more of the surfaces, there may bea corresponding automatically generated common direction handle which isa vector, V, oriented from p, wherein V is perpendicular to the profilehandle of the profile P, this profile lying in the common plane formedby the profile handles for the other profiles also having the point p.Further note, the user interface supports allowing the user to eitherdisplay or not display the common direction handles.

Note that it is not necessary to have profile handles and isoclinehandles that match for profiles that adjoin at a common marker in orderto achieve smooth surfaces there, only that they all lie in a commonplane. In FIG. 18, there are three profile curves 404, 408 and 412 forthe surfaces 416 and 418. Each of the three profile curves meets at theprofile marker 420, and each of the profiles has a correspondingisocline ribbon 424 (for profile 404), 428 (for profile 408), and 432(for profile 412). Additionally, the profile and isocline handlesassociated with the profiles 404, 408 and 412 and the marker 420 are:

-   -   (i) profile handle 436 and isocline handle 440 for profile 404;    -   (ii) profile handle 444 and isocline handle 448 for profile 408;    -   (iii) profile handle 452 and isocline handle 456 for profile        412.        Thus, if the profile and isocline handles 436, 440, 444, 448,        452 and 456 all lie within the plane 460 (indicated by the        dashed rectangular portion), then the surfaces 416 and 418        smoothly join at the marker 420.

At any marker, two isocline ribbons are likely to meet in the way thattwo profiles may meet, that is, two isocline ribbons may have a commonisocline handle as an edge for each of the ribbons.

To achieve tangent plane continuity between different (blended) surfaceregions S₁ and S₂ (FIG. 43) joined by a composite profile, not only istangent continuity across profiles P₁ and P₂ needed, but tangentcontinuity between adjacent ribbons R₁and R₂ is also needed. That is,for the profiles P₁ (between markers 2010 and 2014) and P₂(betweenmarkers 2014 and 2018), the respective ribbons R₁ and R₂, when thoughtof as surfaces, must be tangent plane continuous, and share a commonisocline handle 2022. Note that in most cases, tangent continuitybetween ribbons is equivalent to tangent continuity between profiles andtangent continuity between ribbon boundaries that is required for smoothtransitions across surface patch boundaries. Moreover, the userinterface of the present invention provides techniques for assuringtangent plane continuity between ribbon boundaries wherein thesetechniques are substantially identical to those used for assuringtangent plane continuity between profiles. Thus, the present inventioncan provide tangent plane continuity between adjacent surfaces generatedfrom isocline ribbons according to the present invention.

In some circumstances, it is possible to break the continuity ofcomposite ribbons intentionally, thereby causing a crease across thesurface generated from the ribbons wherein the crease does notcorrespond with a coincident profile along the crease. However, in someembodiments of the present invention, an “implicit profile” can becreated that is coincident with the crease.

4.5.3. Curvature Continuity

The visual quality of a surface depends not only on tangent planecontinuity, but also on higher order derivatives. A user can be acutelysensitive to discontinuous changes in surface curvature, especially ifthe surface is rendered with specular highlights or reflected texturemappings, which is common in simulating realistic scenes. The user mayperceive a distracting visual artifact known as a “Mach band.”Accordingly, raising the order of continuity between transitions to thatof curvature continuity ameliorates this.

Analysis has shown that the curvature of the surface defined by Formula(1) or Formula (4) depends on the second derivatives of the B_(i) andthe S_(i). The dependencies of the B_(i) are non-trivial and it isadvantageous to choose the blending functions so that their secondderivatives are zero and let the surface functions S_(i) determine thecurvature. The cosine squared function of Section 2.1 fulfills thiscondition. There also exist certain quintic polynomials that aresatisfactory. For example, the polynomial B₁(x) of Formulas (3.1).

If the curvature of a blended surface generated from Formula (1) orFormula (4) depends only on the S_(i) (e.g., B_(i)″=0), it is thenpossible to raise the curvature order between the bounding surfacepatches S_(i) analogous to the methods in the previous section forachieving tangent continuity. To do this, simply define thecorresponding profiles and isocline handles so they match in theirsecond derivative at each marker along the profile boundary. Note,however, that each profile handle may be considered as a linear functionof one parameter and therefore has a zero second derivative. Thuscurvature continuity is achieved; albeit by making the curvature acrossthe profile “flat,” i.e., zero. This is useful at points where there isan inflection point on the profile, but can be undesirable elsewhere. Torectify this situation, the linear handles may be replaced with curvedribs, such as parabolic arcs. Accordingly, the handles now become arcs,and at the markers, the curvature is made to match that of the givenarc.

By extending the concept of providing a nonzero curvature to allhandles, e.g., profile, isocline and boundary handles, along with thezero second derivatives of the blending functions and the effects of theMach banding can be mollified.

4.5.4. G1 Continuity Using Roll, Yaw and Magnitude Representations.

The present invention also provides a user interface method to specifyhandle vectors (e.g., isocline handles) relative to a correspondingprofile curve, wherein G1 continuity (as defined in the Definitionshereinabove) between surfaces joined together by the profile is assured.This method, which is denoted herein as the roll-yaw method, specifies avector V in terms of three scalar terms called roll, yaw and mag(magnitude), wherein roll and yaw are determined at a point P on a curveusing the tangent vector T at the point P, and a vector N normal to thecurve at the point P. The yaw component of the vector V represents theangular deviation from T at P. For instance, if the vector V is in adirection perpendicular to T, the yaw value (in at least one framework)is 0°, and if the vector V at P is in the same direction as T, then theyaw value is 90°. Regarding the roll component of the vector V, thisscalar represents the amount of angular rotation about T as the axis ofrotation, and wherein the baseline axis for measuring the angle is thevector N at P. Accordingly, the vector N represents 0° of roll and therotational range extends from −180° to 180° using the right-hand rule,as one skilled in the art will understand. Regarding the magnitudecomponent of vector V, this is simply the length of the vector V. Notethat any vector expressed in terms of three-dimensional Cartesiancoordinates can be transformed one-for-one into the roll, yaw, magnotation for a given T and N.

Note that the vector N may be selected from among vectors in the planenormal to T. However, this does not precisely define N. Thus, severalmethods may be used to define N. A first such method for defining N issimply to choose a constant vector VC and then determine N by thefollowing equation: N=T×VC. This method, however, produces an undefinedvalue for N when T and VC are colinear. To provide values for N wherethis equation yields a zero vector, N can be approximated in atopological neighborhood of the colinearity. Alternatively, in a secondmethod of generating N, the Frenet-Serrat frame of the underlying curvemay be chosen, as one skilled in the art will understand. However, theFrenet-Serrat frame may be discontinuous at inflection points along thecurve. Accordingly, the present invention provides a method for creatinga minimally rotating reference frame for a complex (i.e.,three-dimensional) curve that obviates difficulties in defining thevector N regardless of the orientation or shape of the curve and itstangent vector T.

As previously mentioned the roll-yaw method provides a novel way toachieve G1 continuity across a profile. As an example, consider thegeometry illustrated in FIG. 44, wherein a profile P along with left andright isocline ribbons LR and RR are shown. Each of the isocline ribbonsLR and RR has two corresponding isocline handles at its ends, i.e., HL1and HL2 for LR, and HR1 and HR2 for RR. Assuming the profile P endpointshave handles denoted HR1 and HR2, for any point pp on the profile,continuity across the profile for surfaces bounded thereby is determinedby the interpolated isocline values IL and IR. Further, IR isinterpolated (according to the techniques of the present invention) fromHR1 and HR2, and IL is interpolated from HL1 and HL2. Thus, one skilledin the art will understand that for G1 continuity across the profile P,IL and IR must at least be in opposite (colinear) directions. Further,it can be shown by one skilled in the art that if IR and IL are formedusing a cubic Hermite interpolation between HL1 and HL2 for IL, and, HR1and HR2 for IR that the conditions for G1 continuity are that HL1 andHR1 must be equal and opposite vectors. Further, the same must be truefor HL2 and HR2. However, if instead of interpolating the isoclinevalues IL and IR in Cartesian space, the interpolation is performed in(roll, yaw, mag) space, G1 continuity can be achieved with a less strictcondition, namely, that the roll value of HL1 and HL2 must be the same.Accordingly, this is equivalent to saying that HL1, HR1 and HR1 must beno more than coplanar (with the same being true for HL2, HR2 and HR2) inorder to guarantee G1 continuity everywhere on the profile P. Further,note that similar conditions may be imposed if the isocline handles arecurved rather than straight. In particular, tangent vectors to theisocline handles at their common points with the profile P may be usedin place of any corresponding isocline handle vector represented in FIG.44. Thus, as one skilled in the art will appreciate, computational stepscan be provided that embody the roll-yaw method for, if necessary,converting from Cartesian vectors to roll, yaw, mag vectors, and thenassuring that the above described coplanar constraint is satisfied forguaranteeing that surfaces are G1 continuous across the profile P.

4.6. Embedding Models Within Models

The present invention allows parts of a surface bounded by profiles tobe designed separately from one another. For example, a triangularportion of a surface may be designed as a free standing surface model.That is, a designer may add profiles and isocline ribbons as desireduntil a satisfactory design of the model is obtained (using barycentricmappings as one skilled in the art will understand). Afterwards thispiece may be distorted, rotated and fit into a triangular portion ofanother model. Hence, a finely detailed model may be designed andembedded into another model. By maintaining links this process can beused for level of detail management. That is, for example, when themodel is viewed from a distance, the detailed portion is unneeded fordisplay, but as the viewer moves closer the embedded object is linked infor the extra detail it affords. Two examples of types of embeddingsfollow in the next subsections.

4.6.1. A Rounderizing Technique

Referring to FIG. 19, a small blended surface rounds an edge 482 betweentwo intersecting surfaces 484 and 486. This blended surface 480 isblended from the thin surface strips 488 and 490 whose pre-images are a“small” offset from the pre-image of the edge 482 in parameter space.This process is a straightforward application of Formula (1) where thetwo surfaces 484 and 486 are blended using their common parameter space(not shown).

The new surface types lead to new evaluation routines that areespecially efficient in special cases described.

5. Evaluation

We will first consider the evaluation of the two-edge blend, recognizingthat other forms derive from this fundamental form. Because of itsimportance we will recall Formula (1), which isS(u,v)=S ₁(u,v)B ₁(u,v)+S ₂(u,v)B ₁(u,v).  (1)There are both blending functions B_(i) and isocline ribbons S_(i) todetermine when evaluating the surface S. The blending function iscalculated as a univariate function of distance in the parameter space.As discussed in Section 2, the evaluation of the distance functionvaries considerably depending on how complex the pre-image is inparameter space. Once determined, the actual blending value can becalculated by a simple table look up; that is, the blending functionsare tabulated to a sufficiently high resolution and stored in memorywhere they can be indexed by the input variable. Consider the functionof B₁(x) of Formula (3.1). Evaluate this function at x=0, 0.01, 0.02, .. . , 0.99, and 1. These 1001 values are stored as an array. When apoint X is given, it is used to locate the nearest point in the array,e.g., between 0.52 and 0.53. Subsequently, B(0.52) or B(0.53) are usedas the function value.

There are many techniques that may apply based on what the distance andisocline ribbon functions are. The present discussion is focused on amethod that assumes a simple model computationally, but nevertheless,retains considerable design flexibility. The isocline ribbons 508 (S₁)and 516 (S2) will be given as in FIG. 20. These are parameterized from 0to 1 in both u and v parameters. For each fixed value of v along theprofile line 504, if the corresponding picket on isocline ribbon 508 isa straight line segment (e.g. line segment 512), the isocline ribbon isa ruled surface as one skilled in the art will understand. Accordingly,the parameter u provides a distance-like measurement along the rulingwhere the point (u,v) is found. Assume that each of the isocline ribbons508 and 516 are ruled surfaces. Further assume that the pre-image ofeach of the profiles 504 and 506 in parameter space are the profilesthemselves and the distance-like measurement is the parametric u valueof a point (u,v₀) on the v₀ ruling of the isocline. Because the isoclineribbons 508 and 516 are ruled surfaces, for the constant v₀ parameter wecan scan out a set of equidistant points along line segments 512 and 520by simply adding the appropriate offset vector to the previous value.The initial value is S_(i)(0,v₀). The offset vector is obtained asT ₀ =[S ₁(1,v ₀)−S _(i)(0,v ₀)]/n,  (10)where n is the number of points desired on the ruling line to scan fromone isocline ribbon (pre-image) edge to the opposing other edge.

If we impose the restriction that the blending functions are a partitionof unity, i.e., B₁=1−B₂, which is desirable from a design perspective,then the Formula (1) yieldsS(u, v)=[S ₁ (u, v)−S ₂ (u, v)]·B ₁ (u, v)+S ₂ (u, v)  (11)

In one embodiment, this form and with the previous simplifications, itis seen that each point requires three vector adds (for S₁, S₂ and the“+”), one table look up (for B₁(u,v)) and one scalar multiply. This isafter initialization which consists of finding each S_(i)(0,v) andcomputing T₀, the offset vector (using Formula (10)). To scan out a setof points on S, one simply increments through the parameter v, and thencomputes points along the rulings in u.

In the case of the defined four-edge surface (as in Section 4.6.1), someS_(i) are as the two edge case above, but the others blendlongitudinally across the ribbon first. Specifically, in FIG. 34 thev-loft case is the same as FIG. 35 with re-labeling, while the u-loft ofFIG. 34 is a horizontal blend of isocline ribbons. The four-edge surfaceresults from the barycentric blend of all four.

In FIG. 33, four profile curves P₁₁, P₁₂, P₂₁ and P₂₂ are shown betweenwhich a surface is desired to be placed. In FIG. 33, the profiles P₁₁and P₁₂ together with their corresponding respective isocline ribbonsR₁₁ and R₁₂, are used to create the blended surface S₁.

While S₁ evaluates exactly as the two-edge case already described, thegeneration of S₂ (FIG. 35) differs because the u and v parameters arereversed. In this case, the straight line segment on the isoclines R₂₁and R₂₂ corresponds to fixing u and scanning in v; a situation which isincompatible to rapid scanning. However, it is desirable to fix just oneparameter and scan the other in both FIGS. 34 and 35. In one embodiment,this can be resolved by defining the isoclines R₂₁ and R₂₂ differently.That is, each such ribbon becomes a blend of two ruled surfaces definedby simple user inputs. For example, consider isocline ribbon R₂₁. It maybe defined by blending two bilinear surfaces 1950 and 1952 in a mannersimilar to the surface generation techniques described in section 2.2and illustrated in FIGS. 37 through 39. That is, the edges of thebilinear surfaces 1950 and 1952 that are tangent on the profile P₂₁ arethe profile handles 1956 and 1960; the boundary handles 1964 and 1968are tangent to the ribbon boundary 1972 and form the opposite edges ofthe bilinear surfaces. The other two line segments 1976 and 1980 areuser inputs.

It is now possible to fix v in the second (u-loft) as well and scan byadding a single vector offset. This effort produces points on theisocline ribbons, S₂₁ and S₂₂, each at the identical cost of producingpoints on the v-loft. Additionally, we must blend the new points tocompute the point on S₂. In operation counts, there are, therefore,eleven vector additions, five scalar multiplies and one table look-up.The additions include three for the v-loft, three each for the u-loftisoclines, one for blending these isoclines and one for blending the twolofts.

For the general N-sided surface it is first necessary to compute adistance on each ribbon. The parameters are calculated using the N-sidedparameterization technique from Section 2.2. These distances are thenplugged into the blending functions of Formula (6). They are adjusted sothey vary from 0 to 1.

The parameters for the ribbons must be set from the distance so given.That is, one parameter will be the distance (from the profile). Theother parameter can be deduced by determining where the parameter lineof FIG. 12 crosses the edge of the N-sided polygon. It is assumed thatthe polygon has edges of length 1. After these parameters are figuredthen Formula (4) has all constituents needed for calculation.

6. Applications

The present invention may be used in a large number of computationaldesign domains. In particular, the following list provides briefdescriptions of some of the areas where the present invention can beapplied.

6.1 Container Design

Free-form design of containers such as bottles has been heretoforenon-intuitive and tedious. The present invention alleviates thesedrawbacks.

6.2 Automotive Design

In the automotive industry, the present invention can be used for autobody design as well as for auto component design. In particular, theease with which deformations of parts and contours can be performed withthe present invention allows for straightforward deformation ofcomponents and recesses so that the fitting of components in particularrecesses is more easily designed.

6.3 Aerospace

The present invention provides high precision trimming and surfacepatching operations which are required by the aerospace industry.

6.4 Shipbuilding

Unique to the shipbuilding industry is the need for the design of shiphulls and propellers. Designs of both hulls and propellers may be drivenby the physics of the constraints related to water flow. Thesatisfaction of such constraints can be incorporated into the presentinvention.

6.5 Traditional CAD/CAM Applications

Applications for the design of engines, piping layouts and sheet metalproducts typically require trimming and blending capabilities. Thus,since the present invention is particularly efficient at providing suchoperations as well as providing easy deformations of surfaces, itseffectiveness in these areas may be of particular merit.

6.6 Other Applications

The following is a list of other areas where the present invention maybe used for computational design. These are: home electronic andappliance design, plastic injection mold design, tool and die design,toy design, geological modeling, geographical modeling, mining design,art and entertainment, animation, sculpture, fluid dynamics,meteorology, heat flow, electromagnetics, plastic surgery, burn masks,orthodontics, prosthetics, clothing design, shoe design, architecturaldesign, virtual reality design, scientific visualization of data,geometric models for training personnel (e.g., medical training).

The foregoing discussion of the invention has been presented forpurposes of illustration and description. Further, the description isnot intended to limit the invention to the form disclosed herein.Consequently, variations and modifications commensurate with the aboveteachings, within the skill and knowledge of the relevant art, arewithin the scope of the present invention. The embodiment describedhereinabove is further intended to explain the best mode presently knownof practicing the invention and to enable others skilled in the art toutilize the invention as such, or in other embodiments, and with thevarious modifications required by their particular application or usesof the invention. It is intended that the appended claims be construedto include alternative embodiments to the extent permitted by the priorart.

1. A method for modifying a representation of a surface on acomputational system, comprising graphically displaying a particularsurface having first and second curves provided thereon; activating, bya user of the computational system, a user interface technique fordeforming said particular surface, wherein the following steps A-1through A-3 are performed: (A-1) first determining points on a firstgeometric object representation, wherein said first geometric objectrepresentation represents first data used in evaluating a desiredcontour of said particular surface at a majority of points on said firstcurve that are representable by said computational system; (A-2) seconddetermining points on a second geometric object representation, whereinsaid second geometric object representation represents second data usedin evaluating a desired contour of said particular surface at a majorityof points on said second curve that are representable by saidcomputational system; (A-3) generating a modified version of saidparticular surface, wherein there are a plurality of new points on saidmodified version that are not on said particular surface, and each saidnew point is determined as a function of: (a) at least one pointobtained from said first geometric object representation, and (b) atleast one point obtained from said second geometric objectrepresentation; and graphically displaying said modified version; andwherein said generating step includes computing, for each of one or moreof said new points, a weighted sum of at least one point on each of saidfirst and second geometric object representations.
 2. The method asclaimed in claim 1, wherein at least one of said first and secondgeometric object representations includes a representation of a surface.3. The method as claimed in claim 1, wherein said computing stepincludes establishing for each of one or more of said new points, one ormore corresponding weights of said weighted sum, wherein each saidweight, w, scales a corresponding point, p_(w), of one of said first andsecond geometric object representations.
 4. The method as claimed inclaim 3, wherein said establishing step includes determining, for eachpoint q of a set Q of said new points, and at least one weight, w_(q),of said corresponding weights for said new point q, a distance-likemeasurement D_(w) _(q) that is dependent upon: (a) a pre-imageparameterization of said new point q, and (b) a pre-imageparameterization of a point s_(q) that is on both said modified version,and said geometric object representation containing said point, p_(w)_(q) , for said weight w_(q).
 5. The method as claimed in claim 4,wherein for q₁ and q₂ of said set Q, when said pre-imageparameterization of q₁ is closer to a pre-image parameterization ofs_(q) ₁ than said pre-image parameterization of q₂ is to a pre-imageparameterization of S_(q) ₂ then D_(w_(q₁)) ≤ D_(w_(q₂)).
 6. A methodfor deforming a particular surface, comprising: providing a particularsurface; first obtaining first data for defining a first geometricobject having a dimension greater than or equal to two, wherein saidfirst geometric object includes a first part that substantiallycoincides with a first portion of said particular surface; firstdetermining first points of said first geometric object, said first dataused in determining said first points, wherein at least some of saidfirst points are spaced apart from said particular surface; secondobtaining second data for defining a second geometric object, whereinsaid second geometric object includes a second part that substantiallycoincides with a second portion of said particular surface; seconddetermining second points of said second geometric object, said seconddata used in determining said second points, wherein at least some ofsaid second points are spaced apart from said particular surface;generating a modified version of said particular surface, wherein (i)and (ii) following are satisfied: (i) said modified version includessaid first and second portions; and (ii) there are a plurality of newpoints on said modified version that are not on said particular surface,and each said new point is determined as a function of (a) and (b)following: (a) for at least one point, p₁, from the at least some ofsaid first points, a distance-like value indicative of a spacingbetween: a position identified by a location of p₁, and said first part,and (b) for at least one point, p₂, from the at least some of saidsecond points, a distance-like value indicative of a spacing between: aposition identified by a location of p₂, and said second part; andgraphically displaying said modified version; wherein said generatingstep includes determining, for each of one or more of said new points, aweighted sum obtained using at least one point from each of said firstpoints and said second points.
 7. The method as claimed in claim 6,wherein at least one of said first and second data includes arepresentation of a surface.
 8. The method as claimed in claim 6,wherein said determining step includes establishing for each of one ormore of said new points, one or more corresponding weights of saidweighted sum, wherein each said weight, w, scales a corresponding point,p_(w), from one of said first points and said second points.
 9. Themethod as claimed in claim 8, wherein said establishing step includesdetermining, for each point q of a set Q of said new points, and atleast one weight, w_(q), of said corresponding weights for said newpoint q, a distance-like measurement D_(w) _(q) that is dependent upon:(a) a pre-image parameterization of said new point q, and (b) apre-image parameterization of a point s_(q) that is on both: (1) saidmodified version, and (2) one of said first and second geometric objectscontaining said corresponding point, p_(w) _(q) for said weight w_(q).10. The method as claimed in claim 9, wherein for q₁ and q₂ of said setQ, when said pre-image parameterization of q₁ is closer to a pre-imageparameterization of s_(q) ₁ than said pre-image parameterization of q₂is to a pre-image parameterization of s_(q) ₂ thenD_(w_(q₁)) ≤ D_(w_(q₂)).
 11. A method for providing a representation ofa surface on a computational system, comprising: providing or using acomputational system for performing steps (1) through (4) following: (1)first obtaining first shape data indicative of a shape of at least afirst portion of a first surface; (2) first determining, using saidfirst shape data, at least a collection (C) of representations ofgeometric points, wherein the collection C is indicative of a firstgeometric object having a dimension greater than or equal to two, andwherein (A-1) and (A-2) following hold: (A-1) each of the pointrepresentations of C includes geometric position data that is invariantbetween graphical views, and, in at least one graphical view of thepoint representations, the point representations are distinct from saidfirst portion; (A-2) each point representation of the collection C isdetermined using said first shape data; (3) generating a representation(R) of said first surface, wherein for at least a subcollection ofpoints of R, the subcollection being representative of a surface portionof said first surface, each point q of the subcollection, is determinedby performing the following steps (B-1) through (B-3): (B-1) obtaining apoint representation pq from the collection C; (B-2) determiningcloseness data, d_(q), indicative of a closeness between p_(q) and acorresponding point representation p_(s) of said first shape data,wherein p_(s) is used in determining p_(q); and (B-3) using saidcloseness data d_(q) and the point representation p_(q) in determiningq; wherein as instances of the point representation p_(q) becomeprogressively closer to their corresponding point representations p_(s),the corresponding instances of closeness data d_(q) are used fordetermining corresponding instances of q such that said correspondinginstances of q become progressively closer to their respectivecorresponding p_(q) instances; and (4) graphically displaying saidrepresentation R of said first surface.
 12. The method of claim 11,wherein for substantially all points p of said first geometric object,the graphical view invariant geometric position data for p is determinedusing a corresponding geometric computation for the first geometricobject, wherein said corresponding geometric computation receives asinput at least two independently varying parameters, wherein: (a) whenvalues are provided for the parameters such that a first of theparameters has a fixed value and a second of the parameters is assignedvalues over a range of values, a first set of geometric position datafor the first geometric object is determined, and (b) when values areprovided for the parameters such that a second of the parameters has afixed value and the first parameter is assigned values over a range ofvalues, a second set of geometric position data for the first geometricobject is determined; wherein said first and second sets of points aredifferent; and wherein said step of first determining includesdetermining the geometric position data of p_(q) in the collection C byusing said corresponding geometric computation with values for the atleast two independent parameters corresponding to p_(q).
 13. The methodof claim 11, wherein said step of first determining includes determiningthe geometric position data of p_(q) by activating a geometriccomputation with values for at least two independent parameterscorresponding to p_(q), wherein said geometric computation is determinedusing said first shape data; and wherein: (a) when values are providedfor the parameters such that a first of the parameters has a fixed valueand a second of the parameters is assigned values over a range ofvalues, a first set of geometric position data for the first geometricobject is determined, and (b) when values are provided for theparameters such that a second of the parameters has a fixed value andthe first parameter is assigned values over a range of values, a secondset of points of the first geometric object are determined; wherein saidfirst and second sets of geometric position data are different.
 14. Themethod of claim 11, wherein said first shape data includes tangent dataindicative of a tangent to said first portion of the first surface. 15.The method of claim 14, wherein said tangent is tangent to said firstgeometric object.
 16. The method of claim 11, wherein said surfaceportion (SP), represented by the subcollection, has a parametricpre-image of at least two independent variables.
 17. The method of claim11, wherein said surface portion (SP), represented by the subcollection,has a non-zero surface area.
 18. The method of claim 11, whereinsubstantially all points of said first shape data are included in saidfirst geometric object.
 19. The method of claim 11, whereinsubstantially all points of said first shape data are points of saidfirst portion.
 20. The method of claim 11, wherein substantially allpoints of said first portion are points of said first shape data. 21.The method of claim 11, wherein for at least some of the points p_(q),the corresponding point p_(s) also corresponds with a point of saidfirst portion.
 22. The method of claim 11, wherein said graphicaldisplay of said representation R includes said first portion.
 23. Themethod of claim 11, wherein said first portion consists essentially of aone dimensional geometric object.
 24. The method of claim 23 whereinsaid first geometric entity is a curve.
 25. The method of claim 11,wherein R includes a plurality of first surface points that are notrepresentative of locations of points of said first geometric object.26. The method of claim 11, wherein at least some instances of saidcloseness data d_(q) are determined from using parametric pre-images ofcorresponding instances of p_(s) and p_(q).
 27. The method of claim 11,wherein at least most of the point representations of C are not within apredetermined distance of said first geometric entity, wherein thepredetermined distance is sufficiently small so that a value that is notlarger than said predetermined distance is considered substantiallyequivalent to zero.
 28. The method of claim 11, wherein said firstgeometric entity includes a representation that is invariant betweengraphical views.
 29. The method of claim 11, wherein said first shapedata includes geometric data that is invariant between graphical views.30. The method of claim 11, wherein said step of first obtainingincludes using points of said first portion for obtaining one or more oforientation and position information included in said first shape data,wherein said orientation information is used for defining an orientationof said first geometric object, and said position information is usedfor defining a position of said first geometric object.
 31. The methodof claim 11, wherein for each geometrical view of said first geometricobject, when said first portion is also displayed, the display of saidfirst portion is coincident with a display of said first geometricobject.
 32. The method of claim 11, wherein said computational systemfurther performs the following steps: selecting a particular surface fordeforming; deforming said particular surface for obtaining a differentcontour, and thereby obtaining said first surface as a deformed versionof said particular surface; wherein said step of deforming includesperforming said steps first obtaining, first determining, andgenerating.
 33. The method of claim 32, wherein for at least a portionof said particular surface, there is a corresponding portion of saidfirst surface that effectively coincides therewith.
 34. The method ofclaim 32, wherein said first surface effectively includes at least aportion of said first geometric object.
 35. The method of claim 34,wherein said particular surface effectively includes the portion of saidfirst geometric object.
 36. The method of claim 35, wherein the portionof said first geometric object is no more than a predetermined distancefrom said particular surface, wherein the predetermined distance issufficiently small so that a value that is not larger than saidpredetermined distance is considered substantially equivalent to zero.37. The method of claim 32, wherein said computational system furtherperforms the following steps: receiving a substantially continuous timeseries of input change requests indicative of requested deformations tosaid particular surface; and repeatedly performing said deforming stepas a response to the change requests.
 38. The method of claim 37,wherein said step of repeatedly performing includes interativelyperforming said deforming step during said step of receiving so that theparticular surface is perceived as being deformed substantially in realtime with the receiving of said change requests.
 39. The method of claim38, wherein said first geometric object is effectively coincident withsaid particular surface at said first geometric object.
 40. The methodof claim 32, wherein said computational system further performs a stepof: second determining of at least a collection (C₂) of representationsof geometric points indicative of a second geometric object differentfrom said first geometric object, wherein the point representations ofC₂ are invariant between graphical views, and in at least one graphicalview of the point representations of C₂, said point representations aredisplayed distinct from said first portion; said step of generatingfurther including, for determining each point r of at least some pointsof R, the following steps (C-1) through (C-3): (C-1) obtaining a pointrepresentation p_(r) from the collection C₂; (C-2) determining closenessdata, d_(r), indicative of a closeness between p_(r) and a correspondingpoint representation p_(t) for a second shape of at least a particularportion of said second geometric object, wherein p_(t) is used indetermining p_(r); and (C-3) using said closeness data d_(r) and thepoint representation p_(r) in determining r.
 41. The method of claim 40,wherein for at least one point of R, said steps (B-1), (B-2), (B-3),(C-1), (C-2) and (C-3) are performed for determining the at least onepoint of R, wherein q and r are identical to the at least one point ofR.
 42. The method of claim 41, wherein said steps (B-3) and (C-3) areincluded in a step of performing a weighted summation.
 43. The method ofclaim 40, wherein for a second portion of said particular surface, thereis a second corresponding portion of said first surface that effectivelycoincides said second portion of said particular surface; and saiddifferent contour approaches a second shape, different from said shape,when points of said first surface approach said second geometric object.44. The method of claim 40, wherein said second geometric object has adimension greater than or equal to two, and wherein said secondgeometric object is defined by second data that represents geometricdata that is invariant between graphical views of said second geometricobject.
 45. The method of claim 11, wherein for determining at least oneinstance of q, said step of determining the corresponding closeness datad_(q) includes performing a stellating process.
 46. The method of claim11, wherein said step (B-2) of determining the closeness data d_(q)includes evaluating a distance-like function for obtaining adistance-like measurement between a pre-image of the correspondinginstance of p_(q) and a pre-image of the corresponding instance ofp_(s), wherein said distance-like function satisfies the condition thatas the pre-images of instances of the points p_(q) get progressivelycloser to the pre-image of instances of their corresponding pointsp_(s), then said corresponding distance-like measurements becomeprogressively closer to zero.
 47. The method of claim 11, wherein forpoints q₁ and q₂ of said first surface, when: (i) q₁ is determined byperforming instances of (B-1) through (B-3) thereby obtaining: p_(q) ₁for the point representation p_(q), p_(s) ₁ for the corresponding pointrepresentation p_(s), and d_(q) ₁ for the corresponding closeness datad_(q), and (ii) q₂ is determined by performing instances of (B-1)through (B-3) thereby obtaining: p_(q) ₂ for the point representationp_(q), p_(s) ₂ for the corresponding point representation p_(s), andd_(q) ₂ for the corresponding closeness data d_(q), then when d_(q) ₁<d_(q) ₂ , then q₁ is more strongly influenced by p_(q) ₁ than q₂ isinfluenced by p_(q) ₂ .
 48. The method of claim 11, wherein said step(B-3) includes a step of inputting said corresponding closeness datad_(q) to a first blending function (B₁).
 49. The method of claim 48,wherein said step (B-3) includes a step of scaling said pointrepresentation p_(q) by an output from said blending function B₁. 50.The method of claim 48, wherein said blending function B₁ satisfies thefollowing conditions (i) through (iii): (i) B₁(d_(q)) approaches a firstpredetermined value as d_(q) approaches zero; (ii) B₁(d_(q)) approachesa second predetermined value as d_(q) approaches a predeterminednon-zero value; and (iii) B₁ is one or more of: continuous anddifferentiable on a domain including a range of d_(q) effective forcomputing points q of the subcollection.
 51. The method of claim 50,wherein said computational system further performs the steps of: secondobtaining second shape data indicative of a shape of at least a secondportion of the first surface; second determining, using said secondshape data, at least a collection (C₂) of representations of geometricpoints indicative of a second geometric object, wherein at least some ofsaid geometric point representations of C₂ do not effectively coincidewith said second portion; wherein said second geometric object isindicative of a shape of said first surface substantially at the secondportion; wherein for determining at least one of the points q of thesubcollection, the following steps (C-1) through (C-3) are performed:(C-1) obtaining a point representation u_(q) from the collection C₂;(C-2) determining closeness data, dd_(q), indicative of a closenessbetween u_(q) and a corresponding point representation p_(t) of saidsecond shape data; (C-3) using the closeness data, dd_(q) and the pointrepresentation u_(q) in determining q, including a step of inputting thecorresponding closeness data dd_(q) to a blending function B₂, whereinB₂ satisfies the following conditions (i) through (iii): (i) B₂(dd_(q))approaches a third predetermined value as dd_(q) approaches zero; (ii)B₂(dd_(q)) approaches a fourth predetermined value as dd_(q) approachesa predetermined non-zero value; and (iii) B₂ is continuous anddifferentiable on a domain including a range of dd_(q) effective forcomputing said at least one point wherein said step of (B-3) includes astep of using said closeness data dd_(q) and the point representationu_(q) in determining q.
 52. The method of claim 51, wherein q is afunction of:B ₁(d _(q))*p _(q) and B ₂(dd _(q))*u _(q).
 53. The method of claim 50,wherein said computational system further performs the following steps:selecting a particular surface for deforming; deforming said particularsurface for obtaining a different shape, and thereby obtaining saidfirst surface as a deformed version of said particular surface; whereinsaid step of deforming includes performing said steps first obtaining,first determining, and generating; wherein when there is second shapedata indicative of at least a particular portion of the particularsurface such that said particular portion is not deformed in said stepof deforming, then B₁(d_(q)) approaches zero when p_(q) approaches saidparticular portion.
 54. The method of claim 11, wherein saidcomputational system further performs the steps of: second obtainingsecond shape data indicative of a shape of at least a second portion ofthe first surface; second determining, using said second shape data, atleast a collection (C₂) of representations of geometric pointsindicative of a second geometric object, wherein at least some of saidgeometric point representations of C₂ do not effectively coincide withsaid second portion; wherein said second geometric object is indicativeof a shape of said first surface substantially at the second portion;wherein for determining at least one of the points q of thesubcollection, the following steps (C-1) through (C-2) are performed:(C-1) obtaining a point representation u_(q) from the collection C₂;(C-2) determining closeness data, dd_(q), indicative of a closenessbetween u_(q) and a corresponding point representation p_(t) of saidsecond shape data, wherein p_(t) is used in determining u_(q); andwherein said step of (B-3) includes a step of using said closeness datadd_(q) and the point representation u_(q) in determining the at leastone point q of the subcollection.
 55. The method of claim 54, whereinsaid step of (B-3)_determines q using weightings for p_(q) and u_(q).56. The method of claim 54, wherein said step of(B-3)_determines atleast one instance of q as a weighted sum, said weighted sum including aweighted sum of the corresponding p_(q) and the corresponding u_(q.) 57.The method of claim 54, wherein said computational system furtherperforms a step of inputting said corresponding data dd_(q) to ablending function B₂, wherein B₂ satisfies the following conditions: (i)B₂(dd_(q)) approaches a third predetermined value as dd_(q) approacheszero; (ii) B₂(dd_(q)) approaches a fourth predetermined value as dd_(q)approaches a non-zero value; and (iii) B₂ is continuous anddifferentiable on a domain including a range of dd_(q) effective forcomputing the at least one point q of the subcollection.
 58. The methodof claim 57, wherein the at least one point q of the subcollection is afunction of B₂(dd_(q))*u_(q).
 59. The method of claim 57, wherein saidthird predetermined value is effectively zero.
 60. The method of claim11, wherein said function is such that: (i) the first geometric objecteffectively includes the first portion of the first surface; and (ii) acontour of said first surface approaches said shape of the first portionwhen points of said first surface approach said first portion.
 61. Themethod of claim 60, wherein said first geometric object and said firstportion are within 10⁻³ to 10⁻⁶ of one another.
 62. The method of claim60, wherein a distance between said first geometric object and saidfirst portion is settable by a user.
 63. The method of claim 11, whereinsaid step of graphically displaying is performed in an application forone of: designing a container, designing a vehicle body, aerospacetrimming and patching operations, designing a watercraft hull orwatercraft propeller, designing an engine, designing a piping layout,designing a sheet metal product, designing toys, tool and die design,clothing design, shoe design, architectural design, virtual realitydesign.
 64. The method of claim 11 wherein when the instances of q aresuch that instances of their respective said corresponding pointrepresentations p_(q) extend progressively further away from said firstportion, the instances of q become progressively less dependent uponsaid instances of their respective said corresponding pointrepresentations p_(q).
 65. The method of claim 11, wherein saidcomputational system further performs the steps of: receiving anidentification of a geometric entity indicative of having a shape of atleast a second portion of the first surface; second determining, usingsaid geometric entity, at least a collection (C₂) of representations ofgeometric points indicative of a second geometric object, wherein saidpoint representations of C₂ are representative of a surface area of thesecond geometric object, said point representations of C₂ used indetermining a second shape for said first surface; wherein a geometriccomputation determines said first surface so that (C-1) and (C-2)following are satisfied: (C-1) there is a corresponding portion of saidsecond geometric object that effectively coincides with first surface,wherein said corresponding portion substantially has the shape of saidgeometric entity; and (C-2) said second shape of said first surfaceapproaches said shape of said geometric entity when said first surfaceapproaches said corresponding portion.
 66. The method of claim 65,wherein at least one point q of said subcollection is determined using apoint representation p from C₂, and wherein the at least one point q isdetermined by performing the following substeps: (D-1) determiningcorresponding data, d₂, indicative of a closeness between the pointrepresentation p and the corresponding portion; and (D-2) using saidcorresponding data d₂ with the point representation p to determine theat least one point q.
 67. The method of claim 65, wherein said portionof said second geometric object is a curve.
 68. The method of claim 67,wherein said curve effectively overlaps with said geometric entity. 69.The method of claim 65, wherein each of said point representations of C₂includes geometric position data that is invariant between graphicalviews.
 70. The method of claim 69, wherein said second geometric objecthas a dimension greater than or equal to two.
 71. The method of claim65, wherein said first surface extends at least between said firstportion and said second portion.
 72. The method as claimed in claim 65,further including a step of identifying, by a user, one or morereference objects, from a graphical display device, wherein each of saidreference objects is co-located with one of said first and secondportions of said first surface.
 73. The method of claim 72, wherein fora first of the reference objects, the first reference object iseffectively co-located with said first portion, and said first referenceobject is used to determine said first shape data.
 74. The method ofclaim 73, wherein for a second of the reference objects, the secondreference object is effectively co-located with said second portion, andsaid second reference object is used to determine a shape of said firstsurface at said second portion.
 75. The method of claim 72, wherein atleast one of said reference objects is a marker.
 76. The method of claim65, wherein said geometric computation determines points of said firstsurface by applying weightings to data indicative of positionsidentified by locations of points of said first and second geometricobjects.
 77. The method as claimed in claim 76, further including a stepof determining said weightings as outputs of one or more blendingfunctions, wherein there is a first blending function, B₁, fordetermining weights for positions identified by locations of points ofsaid first geometric object, and there is a second blending function,B₂, for determining weights for positions identified by locations ofpoints of said second geometric object.
 78. The method of claim 65,wherein said computational system further performs the following steps:providing one or more additional geometric entities; obtaining, for eachadditional geometric entity (GE) of said one or more additionalgeometric entities, a corresponding geometric object (G_(GE)) having aportion that effectively coincides with at least a portion of theadditional entity GE; wherein each of said corresponding geometricobject G_(GE) is determined using said additional geometric entity GE towhich said corresponding geometric object corresponds; third determiningcorresponding points for each of said corresponding geometric objectsG_(GE), wherein for each of said corresponding geometric objects G_(GE),at least some of said corresponding points do not effectively coincidewith said additional geometric entity GE to which said correspondinggeometric object corresponds, and said corresponding points are used indetermining a corresponding shape for said first surface; wherein saidgeometric computation determines said first surface such that for eachof said corresponding geometric objects G_(GE), (D-1) and (D-2)following are satisfied: (D-1) there is a corresponding portion of saidfirst surface that effectively coincides with the correspondinggeometric object G_(GE); and (D-2) a contour of said first surfaceapproaches said corresponding shape for the corresponding geometricobject G_(GE) when points of said first surface approach said additionalgeometric entity GE to which the corresponding geometric objectcorresponds.
 79. The method of claim 65, wherein said geometriccomputation is progressively less dependent upon said pointrepresentations of C₂ as geometric position data for said pointrepresentations of C₂ extent progressively further away from said secondportion, wherein for each of the point representations of C₂, thegeometric position data the point representation is invariant betweengraphical views.
 80. The method of claim 65, wherein said generatingstep includes computing a weighted sum of geometric positions forobtaining said point, q, of R, wherein said geometric positions areobtained using at least one point from each of said collections C andC₂.
 81. The method as claimed in claim 80, wherein said weighted sumincludes weights wherein at least some of said weights are obtainedusing a corresponding blending function, said blending function having arange of 0 to
 1. 82. The method as claimed in claim 80, wherein saidweighted sum includes weights wherein for the point q, there are one ormore of said weights that are dependent upon a parametric pre-image ofq.
 83. The method of claim 65, wherein for each of the points q, q isdependent upon a corresponding point p₁ from said first geometric objectand upon a corresponding point p₂ from second geometric object.
 84. Themethod of claim 83, wherein when a parametric pre-image of q is closerto a parametric pre-image of said first portion than to a parametricpre-image of said second portion, then q is more dependent upon p₁ thanp₂.
 85. The method of claim 84, wherein at least one object of saidfirst and second geometric objects includes a parametric surface havinga parameterization.
 86. The method of claim 85, wherein said geometriccomputation uses said parameterization for obtaining said first surfaceas a blending to another surface, wherein said at least one object, ofsaid first and second geometric objects, provides a characteristic ofsaid blending.
 87. The method of claim 86, wherein said characteristicis one of: a tangent continuity, a curvature continuity, G1 continuity,differentiability, curvature, and arbitrarily high orders of continuity.88. The method as claimed in claim 11, wherein said computational systemsimultaneously performs the following steps (C-1) and (C-2): (C-1)changing said first shape data in response to a user input of asubstantially continuous time series of change requests for changingsaid first surface into a second surface; (C-2) displaying said secondsurface according to modifications indicative of said change requests;wherein during an inputting of one of said change requests, saiddisplaying step simultaneously performs a substep of graphicallydisplaying, for one or more previously input change requests of saidtime series, corresponding modifications to said first surface.
 89. Themethod as claimed in claim 11, wherein said computational system furtherperforms the following steps: obtaining additional data indicative of achange in shape of said first geometric object; and using saidadditional data to determine a corresponding changed version of saidfirst geometric object; wherein said additional data is invariantbetween different graphical views.
 90. The method as claimed in claim89, wherein said computational system further performs the followingsteps: determining a different collection C′ of representations ofgeometric points using said changed version of said first geometricobject; re-performing said step of generating using said differentcollection C′ in place of said the collection C.
 91. The method asclaimed in claim 11, wherein said computational system further performsthe following steps: obtaining a portion of said first surface as adevelopable surface; constructing a closed curve boundary for saiddevelopable surface, wherein said boundary identifies an interior ofsaid closed curve on said developable surface; wherein said step ofconstructing includes generating said boundary as a profile curve havinga corresponding isocline ribbon for use in deriving a blended surfacehaving said boundary as a boundary for said blended surface; trimmingsaid developable surface to approximately said boundary so thatsubstantially only said interior of said developable surface isgraphically displayed; and graphically attaching a label to saidinterior so that said label substantially covers said interior.
 92. Themethod as claimed in claim 11, wherein said computational system furtherperforms the following steps: changing one of a shape and a orientationof said first geometric object thereby obtaining a changed firstgeometric object; second displaying, to a user, a second display ofanother version of said first surface, wherein said another version isdetermined using said changed first geometric object, and wherein ashape difference between said first surface and said another version isdetermined using a value for said one of the orientation and shapechanged in said step of changing; and wherein the second display of saidanother version includes a point, q₁, derived using said changed firstgeometric object, wherein said point q₁ is not representative of a pointof said changed first geometric object.
 93. A method as claimed in claim92, wherein said computational system further performs the followingstep of :iteratively performing said steps of changing and seconddisplaying, wherein each iteration of said changing step includesinputting, by the user, a next portion of a substantially continuoustime series of change requests for changing said first surface; whereinsaid iteratively performed steps of changing and second displaying areinterleaved so that the user perceives a substantially real timedeformation of said first surface during said continuous time series ofchange requests.
 94. The method as claimed in claim 11, furtherincluding: identifying, by a user, one or more reference objects from adisplay of a graphical display device; wherein said computational systemfurther performs the following steps: determining said first shape datausing of at least some of: (i) through (iii) following: (i) a locationof one of said reference objects, (ii) directional informationindicative of one or more directions from said reference objects, and(iii) a tangent to one of said reference objects; wherein said firstshape data provides geometric information for determining at least oneof a position, orientation, and shape for said first geometric object.95. The method as claimed in claim 94, wherein said computational systemfurther performs the following steps: first generating said firstgeometric object using said first shape data; and graphically displayingsaid first geometric object.
 96. The method as claimed in claim 11,wherein a first portion of said first surface follows a shape of saidfirst geometric object and a second portion of said first surfacefollows a shape of a second geometric object, and wherein points of saidfirst surface are determined by applying weightings to results frompoints on said first and second geometric objects.
 97. The method asclaimed in claim 11, wherein said computational system further performsthe following steps: providing a representation of a curve having anextent along said curve; obtaining, for each of a plurality of points onthe curve extent, corresponding data indicative of a shape of said firstgeometric object in at least a neighborhood of the point; wherein saidfirst geometric object follows a shape of the curve throughout saidextent; determining data D representative of a change in location of atleast one point of said first geometric object, said at least one pointnot being a point on said first surface; modifying a shape of said firstsurface by using said data D, thereby obtaining a second surface; anddisplaying said second surface with said modified shape on a graphicaldisplay.
 98. Data storage for a computer program that performs thefollowing steps: (1) first obtaining first shape data indicative of ashape of at least a first portion of a first surface; (2) firstdetermining, using said first shape data, at least a collection (C) ofrepresentations of geometric points, wherein the collection C isindicative of a first geometric object having a dimension greater thanor equal to two, and wherein (A-1) and (A-2) following hold: (A-1) eachof the point representations of C includes geometric position data thatis invariant between graphical views, and, in at least one graphicalview of the point representations, the point representations aredistinct from said first portion; (A-2) each point representation of thecollection C is determined using said first shape data; (3) generating arepresentation (R) of said first surface, wherein for at least asubcollection of points of R, the subcollection being representative ofa surface portion of said first surface, each point q of thesubcollection, is determined by performing the following steps (B-1)through (B-3): (B-1) obtaining a point representation p_(q) from thecollection C; (B-2) determining closeness data, d_(q), indicative of acloseness between p_(q) and a corresponding point representation p_(s)of said first shape data, wherein p_(s) is used in determining p_(q);and (B-3) using said closeness data d_(q) and the point representationp_(q) in determining q; wherein as instances of the point representationp_(q) become progressively closer to their corresponding pointrepresentations p_(s), the corresponding instances of closeness datad_(q) are used for determining corresponding instances of q such thatsaid corresponding instances of q become progressively closer to theirrespective corresponding p_(q) instances; and (4) graphically displayingsaid representation R of said first surface.